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STRUCTURAL  DYNAMICS:  RECENT  ADVANCES 
Proceedings  of  the  6th  International  Conference 
Volume  II 


Proceedings  of  the  Sixth  International  Conference  on  Recent  Advances  in 
Structural  Dynamics,  held  at  the  Institute  of  Sound  and  Vibration  Research, 
University  of  Southampton,  England,  from  14th  to  17th  July,  1997,  co-sponsored  by 
the  US  Airforce  European  Office  of  Aerospace  Research  and  Development  and  the 
Wright  Laboratories,  Wright  Patterson  Air  Force  Base. 


Edited  by 

N,S.  FERGUSON 

Institute  of  Sound  and  Vibration  Research, 
University  of  Southampton,  Southampton,  UK. 

H.F.  WOLFE 

Wright  Laboratory, 

Wright  Patterson  Air  Force  Base,  Ohio,  USA. 
and 
C  MEI 

Department  of  Aerospace  Engineering, 

Old  Dominion  University,  Norfolk,  Virginia,  USA. 


©  The  Institute  of  Sound  and  Vibration  Research,  University  of  Southampton,  UK. 

ISBN  no.  0-85432-6375 


19970814  055 


VV  T.-  ,  .. 


PREFACE 


The  papers  contained  herein  were  presented  at  the  Sixth  International 
Conference  on  Recent  Advances  in  Structural  Dynamics  held  at  the  Institute  of 
Sound  and  Vibration  Research,  University  of  Southampton,  England  in  July  1997. 
The  conference  was  organised  and  sponsored  by  the  Institute  of  Sound  and 
Vibration  Research  and  co-sponsored  by  the  Wright  Laboratories,  Wright  Patterson 
Air  Force  Base.  We  wish  to  also  thank  the  following  for  their  contribution  to  the 
success  of  the  conference:  the  United  States  Air  Force  European  Office  of  Aerospace 
Research  and  Development.  The  conference  follows  equally  successful  conferences 
on  the  same  topic  held  at  Southampton  in  1980, 1984,1988,1991  and  1994. 

There  are  over  one  hundred  papers  written  by  authors  from  approximately 
20  different  countries,  making  it  a  truly  international  forum.  Many  authors  have 
attended  more  than  one  conference  in  the  series  whilst  others  attended  for  the  first 
time. 


It  is  interesting  to  note  the  change  in  emphasis  of  the  topics  covered. 
Analytical  and  numerical  methods  have  featured  strongly  in  all  the  conferences. 
This  time,  system  identification  and  power  flow  techniques  are  covered  by  even 
more  papers  than  previously.  Also,  there  are  many  contributions  in  the  field  of 
passive  and  active  vibration  control.  Papers  dealing  with  nonlinear  aspects  of 
vibration  continue  to  increase.  These  observations  seem  to  reflect  the  trend  in 
current  research  in  structural  dynamics.  We  therefore  hope  that  the  present  series 
of  International  Conferences  will  play  a  part  in  disseminating  knowledge  in  this 
area. 


We  would  like  to  thank  the  authors,  paper  reviewers  and  session  chairmen 
for  the  part  they  played  in  making  it  a  successful  conference. 

My  personal  thanks  go  to  the  following  individuals  who  willingly  and 
enthusiastically  contributed  to  the  organisation  of  the  event: 

Dr.  H.F.  Wolfe  Wright  Laboratories,  WPAFB,  USA 

Dr.  C.  Mei  Old  Dominion  University,  USA 

Mrs.  M.Z.  Strickland  ISVR,  University  of  Southampton,  UK 

Grateful  thanks  are  also  due  to  many  other  members  of  ISVR  who  contributed  to 
the  success  of  the  event. 


N.S.  Ferguson 


Sixth  International  Conference  on 
Recent  Advances  in  Structural  Dynamics 

Volume  II 


Contents 

Page  No. 

INVITED  PAPER 

R.D.  BLEVINS 

On  random  vibration,  probability  and  fatigue  881 

ACOUSTIC  FATIGUE  I 

58.  J.  LEE  and  K.R.  WENTZ 

Strain  power  spectra  of  a  thermally  buckled  plate  in 

random  vibration  903 

59.  S.A.  RIZZI  and  T.L.  TURNER 

Enhanced  capabilities  of  the  NASA  Langley  thermal 

acoustic  fatigue  apparatus  919 

60.  I.  HOLEHOUSE 

Sonic  fatigue  characteristics  of  high  temperature  materials 

and  structures  for  hypersonic  flight  vehicle  applications  935 

61 .  M.  FERMAN  and  H.F.  WOLFE 

Scaling  concepts  in  random  acoustic  fatigue  953 

ACOUSTIC  FATIGUE  II 

62.  H.F.  WOLFE  and  R.G.  WHITE 

The  development  and  evaluation  of  a  new  multimodal 

acoustic  fatigue  damage  model  969 

63.  B.  BENCHEKCHOU  and  R.G.  WHITE 

Acoustic  fatigue  and  damping  technology  in  composite 
materials  981 

64.  D.  MILLAR 

The  behaviour  of  light  weight  honeycomb  sandwich  panels 

under  acoustic  loading  995 

65.  P.D.  GREEN  and  A.  KILLEY 

Time  domain  dynamic  Finite  Element  modelling  in  acoustic 

fatigue  design  1007 


SYSTEM  IDENTIFICATION  II 


66. 

U.  PRELLS,  A.W.  LEES,  M.I.  FRISWELL  and  M.G.  SMART 
Robust  subsystem  estimation  using  ARMA-modelling  in 
the  frequency  domain 

1027 

67. 

Y.Q.  NI,  J.M.  KO  and  C.W.  WONG 

Mathematical  hysteresis  models  and  their  application  to 
nonlinear  isolation  systems 

1043 

68. 

M.G.  SMART,  M.I.  FRISWELL,  A.W.  LEES  and  U.  PRELLS 

The  identification  of  turbogenerator  foundation  models 
from  run-down  data 

1059 

69. 

C.  OZTURK  and  A.  BAHADIR 

Shell  mode  noise  in  reciprocating  refrigeration 
compressors 

1073 

70. 

T.H.T.  CHAN,  S.S.  LAW  and  T.H.  YUNG 

A  comparative  study  of  moving  force  identification 

1083 

71. 

P.A.  ATKINS  and  J.R.  WRIGHT 

Estimating  the  behaviour  of  a  nonlinear  experimental  multi 
degree  of  freedom  system  using  a  force  appropriation 
approach 

1099 

POWER  FLOW  TECHNIQUES  II 

72. 

R.S.  LANGLEY,  N.S.  BARDELL  and  P.M.  LOASBY 

The  optimal  design  of  near-periodic  structures  to  minimise 
noise  and  vibration  transmission 

1113 

73. 

J.L.  HORNER 

Effects  of  geometric  asymmetry  on  vibrational  power 
transmission  in  frameworks 

1129 

74. 

M.  IWANIEC  and  R.  PANUSZKA 

The  influence  of  the  dissipation  layer  on  energy  flow  in 
plate  connections 

1143 

75. 

H.  DU  and  F.F.  YAP 

Variation  analysis  on  coupling  loss  factor  due  to  the  third 
coupled  subsystem  in  Statistical  Energy  Analysis 

1151 

76. 

S.J.  WALSH  and  R.G.  WHITE 

The  effect  of  curvature  upon  vibrational  power 
transmission  in  beams 

1163 

77. 

S.  CHOI,  M.P.  CASTANIER  and  C.  PIERRE 

A  parameter-based  statistical  energy  method  for  mid¬ 
frequency  vibration  transmission  analysis 

1179 

PASSIVE  AND  ACTIVE  CONTROL  III 


78.  Y.  LEI  and  L.  CHEN 

Research  on  control  law  of  active  suspension  of  seven 

degree  of  freedom  vehicle  model  1195 

79.  M.  AHMADIAN 

Designing  heavy  truck  suspensions  for  reduced  road 
damage  1203 

80.  A.M.  SADRI,  J.R.  WRIGHT  and  A.S.  CHERRY 

Active  vibration  control  of  isotropic  plates  using 
piezoelectric  actuators  1217 

81.  S.M.  KIM  and  M.J.  BRENNAN 

Active  control  of  sound  transmission  into  a  rectangular 
enclosure  using  both  structural  and  acoustic  actuators  1233 

82.  T.J.  SUTTON,  M.E.  JOHNSON  and  S.J.  ELLIOTT 

A  distributed  actuator  for  the  active  control  of  sound 
transmission  through  a  partition  1247 

83.  J.  RO,  A.  A-ALI  and  A.  BAZ 

Control  of  sound  radiation  from  a  fluid-loaded  plate  using 

active  constraining  layer  damping  1257 


ANALYTICAL  METHODS  11 

84.  E.  MANOACH,  G.  DE  PAZ,  K.  KOSTADINOV  and 
F.  MONTOYA 

Dynamic  response  of  single-link  flexible  manipulators  1275 

85.  B.  KANG  and  C.A.  TAN 

Wave  reflection  and  transmission  in  an  axially  strained, 
rotating  Timoshenko  shaft  1291 

86.  Y.  YAMAN 

Analytical  modelling  of  coupled  vibrations  of  elastically 
supported  channels  1329 

87.  R.S.  LANGLEY 

The  response  of  two-dimensional  periodic  structures  to 
harmonic  and  impulsive  point  loading  1345 

NONLINEAR  VIBRATION  III 

88.  H.  OYANG,  J.E.  MOTTERSHEAD,  M.P.  CARTMELL  and 
M.L  FRISWELL 

Stick-slip  motion  of  an  elastic  slider  system  on  a  vibrating 
disc 


1359 


89. 

R.Y.Y.  LEE,  Y.  SHI  and  C.  MEI 

A  Finite  Element  time  domain  multi-mode  method  for 
large  amplitude  free  vibration  of  composite  plates 

1375 

90. 

P.  RIBEIRO  and  M.  PETYT 

Nonlinear  forced  vibration  of  beams  by  the  hierarchical 
Finite  Element  method 

1393 

91. 

K.M.  HSIAO  and  W.Y.  LIN 

Geometrically  nonlinear  dynamic  analysis  of  3-D  beam 

1409 

92. 

R.Y.Y.  LEE,  Y.  SHI  and  C.  MEI 

Nonlinear  response  of  composite  plates  to  harmonic 
excitation  using  the  Finite  Element  time  domain  modal 
method 

1423 

93. 

C.W.S.  TO  and  B.  WANG 

Geometrically  nonlinear  response  analysis  of  laminated 
composite  plates  and  shells 

1437 

ANALYTICAL  METHODS  III 

94. 

R.S.  HWANG,  C.H.J.  FOX  and  S.  McWILLIAM 

The  free,  in-plane  vibration  of  circular  rings  with  small 
thickness  variations 

1457 

95. 

D.J.  GORMAN 

Free  vibration  analysis  of  transverse-shear  deformable 
rectangular  plates  resting  on  uniform  lateral  elastic  edge 
support 

1471 

96. 

R.G.  PARKER  and  C.D.  MOTE,  Jr. 

Wave  equation  eigensolutions  on  asymmetric  domains 

1485 

97. 

A.V.  PESTEREV 

Substructuring  for  symmetric  systems 

1501 

RANDOM  VIBRATION  I 

98. 

G.FUandJ.  PENG 

Anaytical  approach  for  elastically  supported  cantilever 
beam  subjected  to  modulated  filtered  white  noise 

1517 

99. 

S.D.  FASSOIS  and  K.  DENOYER 

Linear  multi-stage  synthesis  of  random  vibration  signals 
from  partial  covariance  information 

1533 

100. 

CW.S.TOand  Z.  CHEN 

First  passage  time  of  multi-degrees  of  freedom  nonlinear 
systems  under  narrow-band  non-stationary  random 
excitations 

1549 

101. 

C.  FLORIS  and  M.C.  SANDRELLI 

Random  response  of  Duffing  oscillator  excited  by  quadratic 
polynomial  of  filtered  Gaussian  noise 

1565 

102. 

S.  McWILUAM 

Extreme  response  analysis  of  non-linear  systems  to  random 
vibration 

1581 

103. 

M.  GHANBARI  and  J.F.  DUNNE 

On  the  use  of  Finite  Element  solutions  of  the  FPK  equation 
for  non-linear  stochastic  oscillator  response 

1597 

RANDOM  VIBRATION  II 

104. 

T.L.  PAEZ,  S.  TUCKER  and  C.  O’GORMAN 

Simulation  of  nonlinear  random  vibrations  using  artificial 
neural  networks 

1613 

105. 

D.Z.  LI  and  Z.C.  FENG 

Dynamic  properties  of  pseudoelastic  shape  memory  alloys 

1629 

106. 

Z.W.  ZHONG  and  C.  MEI 

Investigation  of  the  reduction  in  thermal  deflection  and 
random  response  of  composite  plates  at  elevated 
temperatures  using  shape  memory  alloys 

1641 

SIGNAL  PROCESSING  I 

107. 

M.  FELDMAN  and  S.  BRAUN 

Description  of  non-linear  conservative  SDOF  systems 

1657 

108. 

N.E.  KING  and  K.  WORDEN 

A  rational  polynomial  technique  for  calculating  Hilbert 
transforms 

1669 

109. 

D.M.  LOPES,  J.K.  FIAMMOND  and  P.R.  WHITE 

Fractional  Fourier  transforms  and  their  interpretation 

1685 

SYSTEM  IDENTIFICATION  III 

no.  J.  DICKEY,  G.  MAIDANIK  and  J.M.  D’ARCHANGELO 

Wave  localization  effects  in  dynamic  systems  1701 

111.  P.  YUAN,  Z.F-  WU  and  X.R.  MA 

Estimated  mass  and  stiffness  matrices  of  shear  building 
from  modal  test  data 


1713 


112.  YU.  I.  BOBROVNirSKn 

The  problem  of  expanding  the  vibration  field  from  the 
measurement  surface  to  the  body  of  an  elastic  structure  1719 

113  M.  AMABILI  and  A.  FREGOLENT 

Evaluation  of  the  equivalent  gear  error  by  vibrations  of  a 
spur  gear  pair 


1733 


ON  RANDOM  VIBRATION,  PROBABILITY,  AND  FATIGUE 


R.  D.  Blevins 

Rohr  Inc.,  Mail  Stop  107X 

850  Lagoon  Drive 

Chula  Vista,  California  91910 


ABSTRACT 

Analysis  is  made  to  determine  the  properties  of  a  random  process  consisting  of  the 
sum  of  a  series  of  sine  waves  with  deterministic  amplitudes  and  independent,  random 
phase  angles.  The  probability  density  of  the  series  and  its  peaks  are  found  for  an  arbitrary 
number  of  terms.  These  probability  distributions  are  non-Gaussian.  The  fatigue  resulting 
from  the  random  vibration  is  found  as  a  function  of  the  peak-to-rms  ratio. 


1.  INTRODUCTION 

Vibration  spectra  of  aircraft  components  often  are  dominated  by  a  relatively  small 
number  of  nearly  sinusoidal  peaks  as  shown  in  Figure  1.  The  time  history  of  this  process, 
shown  in  Figure  2,  is  irregular  but  bounded.  The  probability  density  of  the  time  history, 
shown  in  figure  3  only  roughly  approximates  a  Gaussian  distribution  and  it  does  not  exceed 
2.5  standeird  deviations. 

The  time  history  of  displacement  or  stress  of  these  processes  over  a  flight  or  a  take 
off  time  can  be  expressed  as  a  Fourier  series  of  a  finite  number  of  terms  over  the  finite 
sampling  period  T. 

N 

y  =  0,nCOs(u)-ntn  +  <^n),  0  <  in  <  T,  Un  >  0  (l) 

n=l 

Each  frequency  Un  is  a  positive,  non-zero  integer  multiple  of  27r/r.  The  following  model  is 
used  for  the  nature  of  the  Fourier  series:  1)  the  amplitudes  a-n  are  positive  and  deterministic 
in  the  sense  that  they  do  not  vary  much  from  sample  to  sample,  2)  the  phases  (j>n  are  random 
in  the  sense  that  they  vary  from  sample  to  sample,  they  are  equally  likely  to  occur  over 
the  range  -oo  <  0n  <  oo.  This  last  condition  implies  that  the  terms  on  the  right  hand 
side  of  equation  (1)  are  statistically  independent  of  each  other. 

We  can  generate  an  ensemble  of  values  of  the  dependent  variable  Y  by  randomly 
choosing  M  sets  of  N  phase  angles  =  1,2..^),  computing  Y  at  some  ffxed  time 

from  equation  (1),  choosing  another  set  of  phases,  computing  a  second  value  of  Y  and  so 


881 


on  until  we  have  a  statistically  significant  sample  of  M  Y's.  This  random  phase  approach, 
introduced  by  Rayleigh  (1880),  models  a  multi-frequency  processes  where  each  frequency 
component  is  independent  and  whose  power  spectral  density  (PSD)  is  known. 

The  maximum  possible  (peak)  value  of  equation  (1)  is  the  sum  of  the  amplitude  of 
each  term  (recall  >  0).  The  mean  square  of  the  sum  of  independent  sine  waves  is  the 
sum  of  the  mean  squares  of  the  terms. 


N 


^peak  ~  'y  ^ 


n=l 


=  Na,  for  ai=  02=  an  =  CL 


(2a) 

(2&) 


X  N  N 

Yrms  =  ^  f  [Y  anCO«(2wt„/T  +  <Pri)?dtn  =  5 

=  |iVa^,  for  ai  =  02  =  an  =  a 

The  peak-to-rms  ratio  of  the  sum  of  N  mutually  independent  sine  waves  thus  is, 

N  N 


I  rms 


1/2 


n=l  71=1 

=  (2A/')^/^,  for  Oi  =  02  =  an  =  o. 


(3a) 

{Sb) 

(4a) 

(46) 


Equation  (4b)  shows  that  the  peak-to-rms  ratio  for  an  equal  amplitude  series  increases  from 
2^/^  for  a  single  term  (N=l)  and  approaches  infinity  as  the  number  of  terms  N  approaches 
infinity,  as  shown  in  Figure  4.  The  probability  of  Y  is  zero  beyond  the  peak  value.  For 
example,  there  is  no  chance  that  the  sum  of  any  four  {N  =  4)  independent  sinusoidal  terms 
will  be  greater  than  8^/^  =  2.828  times  the  overall  rms  value. 


2.  PROBABILITY  DENSITY  OF  A  SINE  WAVE 

The  probability  density  py  (y)  of  the  random  variable  Y  is  probability  that  the  random 
variable  Y  has  values  within  the  small  range  between  y  and  y  -b  dy,  divided  by  dy.  p(Y) 
has  the  units  of  1/Y.  Consider  single  a  sine  wave  of  amplitude  a^,  circular  frequency  uJri) 
and  phcLse 

Y  =  On  COs{0Jntn  +  0  <  <  277.  (5) 

Y  is  the  dependent  random  variable.  The  independent  random  variables  are  tn  or  <l>n-  The 
probability  density  of  a  sine  wave  for  equal  likely  phases  p((l>n)  =  l/(27r),  or  equally  likely 
times,  p[tn)  =  1/T,  is  (Bennett,  1944;  Rice  1944,  art.  3.10), 


wiy)  = 


77  ^(a^  -  j/2)  1/2^  if  <y  <an\ 


0, 


if  I2/I  >  CLn 


(6) 


882 


The  probability  density  of  the  sine  wave  is  symmetric  about  y  =  0,  i.e.,  pyiv)  ~  PYi~y)) 
it  is  singular  at  y  =  Cn,  and  it  falls  to  zero  for  jyl  greater  than  an  as  shown  in  Figure  5. 

The  characteristic  function  of  a  random  variable  x  is  the  expected  value  of 

C{f)  =  r  (7) 

J  —CO 

and  it  is  also  the  Fourier  transform  of  the  probability  density  function  (Cramer,  1970, 
pp.  24-35;  Sveshnikov,  1965;  with  notation  of  Bendat,  1958).  j  =  is  the  imaginary 
constant.  The  characteristic  function  of  the  sine  wave  is  found  using  equations  (13)  and 
(14)  and  integrating  over  the  range  0  <  X  <  a^.  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  article 
3.753). 


Cn(f)  =  2(7ra„)-‘  T"  cos{27rfY))[l  -  {Y / dY  =  Jo(27r/a„),  (8) 

Jo 

The  characteristic  function  of  a  sine  wave  is  a  Bessel  function  of  the  first  kind  and  zero 
order  (Rice,  1944,  art.  3.16).  Equations  (6)  and  (8)  are  starting  points  for  determining 
the  probability  density  of  the  Fourier  series. 


3.  PROBABILITY  DENSITY  OF  THE  SUM  OF  N  SINE  WAVES 

It  is  possible  to  generate  an  expression  for  the  probability  density  of  Fourier  series 
(equation  l)  with  1,2, 3, to  any  number  of  terms  provided  the  sine  wave  terms  are  mutually 
independent.  This  is  done  with  characteristic  functions.  The  characteristic  function  of  the 
sum  of  N  mutually  independent  random  variables  (Y  =  Xi  -j-  X2  +  -■■i-  Xj\/)  is  the  product 
of  their  characteristic  functions  (Weiss,  1990,  p.22;  Sveshnikov,  pp.  124-129), 

c(/)  =  r  ..  r  e^2-/{^.+^=+- 

7  —00  7—00 

N  ^00  N 

=  n  /  =  n  CM)-  (9) 

The  symbol  11  denotes  product  of  terms.  The  characteristic  function  for  the  sum  of  N 
independent  sine  waves  is  found  from  equations  (8)  and  (9). 


C{f)^ 


n!Li  *^0 (27r/an) ,  unequal  an 
[Jo(27r/a)]^,  ai  ==  a2  =  =  a 


(10) 


The  probability  density  of  Y  is  the  inverse  Fourier  transform  of  its  characteristic  function 
(Sveshnikov,  1968,  p.  129). 


py{y)  =  r  e-^^-fyc{f)df 

7—00 


(11) 


883 


By  substituting  equation  (10)  into  equation  (11)  we  obtain  an  integral  equation  for  the 
probability  density  of  a  N-term  finite  Fourier  series  of  independent  sine  waves  (Barakat, 
1974). 


.oo  N 

Pviy)  =  2  /  cos{2Tryf)  {  TT  Jo(27r/an)  }  df, 

''0  n=l 


iV  =  1,2,3... 


(12) 


If  all  N  terms  of  the  Fourier  series  have  equal  amplitudes  a  =  ai  —  a2  =  On  =  a/sr,  then 
this  simplifies, 


py(y)  =  2  r  cos(2iryf)[Jo{2wfa)fdf,  N  =  1,2,3...  (13) 

Jo 

These  distributions  are  symmetric  about  y  —  0  as  are  all  zero  mean,  sum-of-sine-wave 
distributions.  Figures  5  and  6  show  results  of  numerically  integrating  equations  (45)  and 
(46)  over  interval  /  =  0  to  /  =  15a  using  Mathematica  (Wolfram,  1995). 

Barakat  (1974,  also  see  Weiss,  1994,  p.  25)  found  a  Fourier  series  solution  to  equation 
(45).  He  expanded  the  probability  density  of  the  N  term  sum  in  a  Foui'ier  series  over  the 
finite  interval  -Ly  <  Y  <  Ly  where  Ly  =  ai  +  a2  +  -  +  The  result  for  unequal 
amplitudes  is  , 


\v\<Ly. 

i=L  n=l 

For  equal  amplitudes,  ai  =  a2  =  a,^  =  o,  Ly  =  Na,  and 


i=l 


\y\  <  Na. 


(14) 


(15) 


Figure  6  shows  that  the  Fourier  series  solution  (equation  15)  carried  to  20  terms  to  be 
virtually  identical  to  numerical  integration  of  equation  (13)  and  it  compares  well  with  the 
approximate  solution.  Note  that  theory  requires  py{\yT\  >  Ly)  =  0. 

A  power  series  solution  for  equation  (13)  can  be  found  with  a  technique  used  by  Rice 
(1944,  art.  16)  for  shot  noise  and  by  Cramer  (1970)  who  called  it  an  Edgeworth  series.  The 
Bessel  function  term  in  equation  (13)  is  expressed  as  an  exponent  of  a  logarithm  which  is 
then  expanded  in  a  power  series, 

[Jo(27r/a)]-^  =  ex'p{N  ln[Jo[2'Kaf)\),  (16) 

=  exvi-Nir'^a^f  -  (l/4)iV7r^o'‘/''  -  (l/9)N-K^a^f  +  (n/192)Arx5a®/-)- 

=  +  -1 

Substituting  this  expansion  into  equation  (13)  and  rearranging  gives  a  series  of  integrals, 
which  are  then  solved  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  arts.  3.896,  3.952)  to  give  a 


884 


power  series  for  the  probability  density  of  the  equal-amplitude  N-term  Fourier  series  sum. 


VY{y)  = 


g^iFi[-2,V2,yV(2yA.,)l 


-( 


11 


r(9/2) 


(17) 


192iV3  32iv2^  ^1/2 


)^^iFi[-4,l/2,yV(2i;L.)l  +  -)^  \y\<Na 


'PY{\y\  >  =  0  and  Yrms  is  given  by  equation  (3b).  There  are  two  special  func¬ 

tions  in  equation  (17),  the  gamma  function  T  and  the  confluent  hypergeometric  function 
iFi[n,'y,z].  These  are  defined  by  Gradshteyn,  Ryzhik,  and  Jeffrey  (1994). 


As  N  approaches  infinity,  the  peak-to-rms  (equation  4b)  ratio  approaches  infinity,  and 
equation  (51)  approaches  the  normal  distribution, 

\im  pY{y)  =  ~;^ — .  (18) 

N-*oo  V^Yrms 

as  predicted  by  the  central  limit  theorem  (Cramer,  1970;  Lin,  1976). 


4.  PROBABILITY  DENSITY  OF  PEAKS 

Theories  for  calculating  the  fatigue  damage  from  a  time  history  process  generally 
require  knowledge  of  the  peaks  and  troughs  in  the  time  history.  This  task  is  made  simpler 
if  we  assume  that  the  time  history  is  narrow  band.  If  Y{t)  is  narrow  band  that  is,  that 
each  trajectory  of  Y{t)  which  crosses  zero  has  only  a  single  peak  before  crossing  the  cixis 
again,  then  (1)  the  number  of  peaks  equals  the  number  of  times  the  time  history  crosses 
the  axis  with  positive  slope,  and  (2)  only  positive  peaks  occur  for  Y{t)  >0  and  they  are 
located  at  points  of  zero  slope,  dY{t)ldt  =  0.  Lin  (1967,  p.  304)  gives  expressions  for  the 
expected  number  of  zero  crossings  with  positive  slope  (peaks  above  the  axis)  per  unit  time 
for  a  general,  not  necessarily  narrow  band,  process, 

ElNo+]=  f  ypyy(0,y)<iy  (19) 

Jo 

and  the  probability  density  of  the  peaks  for  a  narrow  band  process. 

=  (20) 

In  order  to  apply  these  expressions,  the  joint  probability  distribution  of  Y  and  Y  must  be 
established.  The  joint  probability  density  function  Pyriy^y)  random  variable 

Y  and  Y  is  the  probability  that  Y  falls  in  the  range  between  y  and  y  +  dy  and  y  falls  in  the 
range  between  y  and  y-\-dy,  divided  by  dydy.  The  derivative  of  the  sine  wave  Y  (equation 
12)  with  respect  to  time  can  be  expressed  in  terms  of  Y , 

dY/dt  =  Y  —  -Gn^n  sin{u)nt  +  0n)  ==  ~  Y^,  \Y\  <  On.  (21) 


885 


The  joint  probability  density  is  the  inverse  Fourier  transform  of  its  characteristic 
function, 

/CO  poo 

/  (30) 

-oo  J—oo 

The  proof  of  equations  (28),  (29),  and  (30)  can  be  found  in  Chandrasekhar  (1943),  Willie 
(1987),  Weiss  and  Shmueli  (1987),  and  Weiss  (1994,  pp.  21-26). 

Since  the  probability  is  symmetric  about  y  —  y  =  0,  Pyriv^v)  ~  Pyy(~S/j“y))  only 
symmetric  terms  survive  the  integration.  Substituting,  equation  (29)  into  equation  (30) 
and  expanding  gives  and  integral  expression  for  the  joint  probability  of  y  and  Y. 

Pyy(2/,y)==  [  [  {JJ  ^o(27ranY^/f  +  /|cj2)}cos(27r/iy)cos(27r/2y)d/i<i/2  (31) 


It  is  also  possible  to  expand  the  joint  probability  of  Y  and  Y  in  as  double  finite  Fourier 
series.  The  result  is: 


..  oo  oo  N  I  .  7 

Pyriv^y)  =  (^)2j}cos(i7ry/Ly)cos(/:7ry/L^) 


aik  =  1,  i,  /c>  0;  1/2,  i  —  Qork  =  0;  1/4,  k  =  0 


(32) 

(33) 


The  expected  number  of  peaks  per  unit  time  and  the  probability  distribution  of  the  narrow 
band  peaks  is  obtained  by  substituting  this  equation  into  equations  (19)  and  (20)  and 
integrating.  The  results  are: 


n—1 


^0  k=0 


TTUr 


Ly  ‘ 


(34) 


y  OO  OO  iV 


:/-)'  + (^P)}sin(i^A/Ly) 
JuY  J^Y 


where 


(  1/8,  i  =  /c  =  0, 

_  J  1/4,  i  >  0,  A:  =  0 

-  1  (l/2)[(-l)'‘-  -  ll/(fc7r)^  i  =  0,fc  >  0, 

I  [(-1)*  -  ll/(fe7r)2,  i>0.fc>0. 


(35) 


(36) 


If  the  frequencies  are  closely  spaced  so  uJn^<^  and  hence  Ly  »  uLy ,  then  one  positive 
peak  is  expected  once  per  cycle, 


£;[iVo+]=a;/(27r) 


(37) 


886 


and  the  probability  density  of  narrow  band  peaks  becomes, 


N 


t=0  fc=0 


71=1 


(38) 


Figure  7  shows  probability  density  of  narrow  band  peaks  for  N=2,3,  and  4  equal  amplitude 
(tti  =  1)  equal  frequency  series  using  equation  (38).  Each  sum  in  equation  (38)  was  carried 
to  40  terms. 

A  power  series  solution  for  equation  (20)  can  be  found  if  all  N  terms  in  the  series  have 
equal  amplitude  and  frequency.  The  result  is 


Pa(A)  = 


Y2 


1  1 
4iV  4ivy2 


A^ 


32Nr,i, 


•h... 


In  the  limit  as  N  becomes  infinite  these  equations  become, 


(39) 


pYi-(y>y)  = 


IttY  Y 

Zr/i  j  rms*^  rms 


(40) 


Pa{A)  = 


y2 

^  rms 


(41) 


Equation  (40)  is  in  agreement  with  an  expression  given  by  Crandall  and  Mark  (1963,  p. 
47)  and  equation  (41) is  the  Rayleigh  distribution. 


Equations  (20),  (35),  (38),  and  (41)  are  conservative  when  applied  to  non-narrow  band 
processes  in  the  sense  that  any  troughs  above  the  axis  (points  with  y  >  0  and  dYjdt  =  0 
but  d^Yldt^  >  0)  are  counted  as  peaks  (Lin,  1967,  p.  304;  Powell,  1958;  Broch,  1963). 


Equations  (35),  (38)  and  (41)  can  provide  probability  distributions  for  peaks  of  narrow 
band  processes  as  a  function  of  the  number  of  sine  waves  from  one  to  infinity  and  thus 
they  model  random  processes  with  peak-to-rms  ratios  from  2^/^  to  infinity. 


5.  FATIGUE  UNDER  RANDOM  LOADING 

Fatigue  tests  are  most  often  made  with  constant-amplitude  sinusoidal  loading.  The 
number  of  cycles  to  failure  is  plotted  versus  the  stress  that  produced  failure  and  the  data 
is  often  fitted  with  an  empirical  expression.  MIL-HDBK-5G  (1994)  uses  the  following 
empirical  expression  to  fit  fatigue  data, 

log  iVy  =  +  B2log{S^  -  S4),  =  5(1  -  R)^K  (42) 

Here  Nf  is  the  cycles  to  failure  during  sinusoidal  loading  that  has  maximum  stress  5  per 
cycle.  R  is  the  ratio  of  maximum  to  minimum  stress  during  a  cycle.  R  =  — 1  is  fully 


887 


reversed  stress  cycling.  Bi  though  B4  are  fitted  parameters.  With  a  little  work,  we  can 
put  this  expression  in  the  form  used  by  Crandall  and  Mark  (1963,  p.  113). 

JV  =  cSJ*-  (43) 

where  Sd  =  5(1  -  -  B4,  c  =  10-®' ,  and  b  =  -82-  For  cycling  in  a  time  history  that 

has  non  constant  amplitude,  Miner-Palmgren  proposed  that  the  accumulated  damage  is 
the  sum  of  the  ratios  of  the  number  of  cycles  at  each  amplitude  to  the  allowing  number  of 
cycles  to  failure  at  that  amplitude  (equations  42  and  43). 

D  =  ^«(Si)/lV^(S,)  (44) 

i 

where  n{Si)  is  the  number  of  cycle  accumulated  at  stress  amplitude  Si  and  Nj  is  the 
number  of  stress  cycles  at  this  amplitude  which  would  cause  failure. 

Following  Miles(1954)  and  Crandall  and  Mark(1963),  the  expected  fractional  damage 
for  a  random  stress  cycling  in  system  with  dominant  cycling  at  frequency  f  in  time  t/,  is 

=  (45) 

where  Pa{S)  is  the  probability  density  of  a  stress  cycle  having  amplitude  S  and  Nf{S)  is 
the  number  of  allowable  cycles  to  failure  at  this  stress.  Failure  under  random  loading  is 
expected  when  the  expected  damage  is  unity.  Setting  =  1  at  time  such  that 

ftd  =  Nd,  the  inverse  of  the  expected  number  of  random  vibration  cycles  to  failure  is 

This  expression  can  be  used  to  create  a  fatigue  curve  for  random  cycling  given  the  proba¬ 
bility  density  of  the  random  stress  cycle  amplitudes  (p^(5))  and  a  fatigue  curve  (equation 
42  with  parameters  Bi  though  B4  and  R)  for  sinusoidal  cycling. 

Substituting  the  probability  density  expression  for  narrow  band  amplitude  (equation 
35)  and  for  the  fatigue  curve(equations  42  or  43)  into  equation  (46)  and  integrating,  we 
obtain  an  expression  for  the  expected  number  of  cycles  to  failure  as  a  function  of  the 
number  of  sine  waves  and  their  amplitudes.  For  N  equal  amplitude  sine  waves  this  is, 


<"->■■  -  I  ^1" 


(47) 


V  (2  +  6)L,(l-B)®=  ' 

+ 


iirBi 


(1  -  R)^Ly 
mBi 


■2’ 

.1  3 


1  T-irr-^  f-**  ^ 

(1  - 


iV(B^(l-B)®’-B4)2, 
ALlil  -  B)2B3  ' 

i^7r^(£„(l  -  B)®°  -  Bif. 
4L|(1  -  H)2®» 


888 


Recall  that  for  this  case  Ly  =  Na,  the  rms  value  is  Y^ms  —  {l/2)Na  and  the  peak- 
to-rms  ratio  is  Peak/Yrms  =  y/2N  (equations  2  though  4).  is  the  generalized 

hypergeometric  function  which  is  a  series  of  polynomials.  It  is  described  by  Gradshteyn, 
Ryzhik,  and  Jeffrey  (1994). 

It  is  also  possible  to  establish  the  fatigue  curve  using  the  Rayleigh  distribution  (equa¬ 
tion  18)  and  the  MIL-HDBK-5  fatigue  curve  (equation  42).  The  result  is 


(2^'^r[l  +  5]((1  - 


-BlF,[l  + 


^  3  Bl 

2’2’2y;2„,(i-ij)2B3« 


(48) 


Bl 


rr2Y,^^{i-RYB,^ 


_orf3  +  il  +  3  Bl 

4  2’  2’  2V;2„,(1  -  ii)2B3 


iFi[..]  is  the  confluent  hypergeometric  function  which  is  described  by  Gradshteyn,  Ryzhik, 
and  Jeffrey  (1994). 

Much  of  the  complexity  of  these  last  two  equations  arises  from  the  term  B4  which  is 
associated  with  an  endurance  limit  in  the  fatigue  equation.  That  is,  equation  (42) predicts 
that  sinusoidal  stress  cycling  with  stress  less  than  54/(1— R) ^=3  produces  no  fatigue  damage. 
If  we  set  B4  =  0  to  set  the  endurance  limit  to  zero,  then  equation  (48)  simplifies  to. 

This  result  for  cycles  to  failure  under  Gaussian  loading  without  an  endurance  limit  is  also 
given  by  Crandall  and  Mark  (1963,  p.  117). 

Equations  (47),  (48)  and  (49)  allow  us  to  compute  the  fatigue  curves  of  a  material 
under  random  loading  from  a  fatigue  curve  generated  under  sinusoidal  loading  (equation 
43)  for  narrow  band  random  processes  with  any  peak-to-rms  ratio  from  2^/^  to  infinity. 


6.  APPLICATION 

Figure  8  is  the  MIL-HDBK-5G  fatigue  curve  for  aluminum  2024-T3  with  a  notch 
factor  of  Kt=4  under  sinusoidal  loading  with  various  R  values.  The  fitted  curve  shown  in 
the  figure,  gives  the  following  parameters  for  equation  (42). 

Bl  =  8.3,  B2  =  -3.30,  Bz  =  0.66,  B4  =  8.4 


889 


The  B2  and  B3  are  dimensionless.  B4  has  the  units  of  ksi,  that  is  thousands  of  psi,  and 
10^^  has  units  of  These  Bi,..B4  are  substituted  into  equations  (43),  (47),  (48), 

and  (49). 

The  fatigue  curves  under  random  loading  are  computed  as  follows,  1)  the  number  of 
sine  waves  N  is  chosen  and  this  fixes  the  peak-to-rms  ratio  from  equation  (3b),  2)  set  of 
values  of  rms  stresses  are  chosen  and  for  each  the  corresponding  sine  waves  amplitudes  are 
computed  using  equation  (3b),  a  =  Srmsy/VN  (note  that  the  peak  stress  much  exceed 
S4=8.5  ksi),  and  3)the  cycles  to  failure  are  calculated  from  equation  (47)  for  finite  peak- 
to-rms  ratios  and  equation  (48)  for  Gaussian  loading  (infinite  peak-to-rms). 

For  single  sine  wave,  the  peak-to-rms  ratio  is  2^/^,  equation  4b,  and  the  fatigue  curve 
interms  of  rms  stress  is  adapted  from  the  empirical  data  fit  (equations  42,  43)  by  substi¬ 
tuting  2^^‘^SrTns  for  the  stress  amplitude. 


Nd  =  c(2^/25.n..(l  -  -  B^r^  (50) 

where  b=  -B2  and  c  =  10-®^  Some  results  are  shown  in  Figure  9  for  R=-l. 

7.  CONCLUSIONS 

Analysis  has  been  made  to  determine  the  properties  of  a  random  process  consisting  of 
the  sum  of  a  series  of  sine  waves  with  deterministic  amplitudes  and  random  phase  angles. 
The  joint  probability  density  of  the  sum  and  its  first  two  derivatives  is  determined.  The 
probability  density  of  the  sum  and  narrow  band  peaks  have  been  found  for  an  arbitrary 
number  of  statistically  independent  sine  wave  terms.  The  fatigue  cycles-to-failure  resulting 
from  these  processes  has  been  found. 

1.  The  peak-to-rms  ratio  of  the  sum  of  mutually  independent  terms  exceeds  unity.  If  ail 
terms  have  the  same  peak  and  rms  values  then  the  peak-to-rms  ratio  of  the  series  sum 
increases  with  the  square  root  of  the  number  of  terms  in  the  series.  The  probability 
of  the  series  sum  is  zero  beyond  a  maximum  value,  equal  to  the  sum  of  the  series 
amplitudes,  and  below  the  minimum  value.  Hence,  he  probability  densities  of  the 
finite  series,  their  peaks,  and  their  envelope  are  non  Gaussian. 

3.  The  formulas  allow  the  direct  calculation  of  the  probability  density  of  the  series  and  its 
peaks  from  its  power  spectra  density  (PSD)  under  the  assumption  that  each  spectral 
component  is  statistically  independent. 

4.  The  fatigue  curves  of  a  material  under  random  loading  with  any  peak-to-rms  ratio 
from  2^/^  to  infinity  can  be  computed  dfrectly  from  the  fatigue  curve  of  the  material 
under  sinusoidal  loading. 


890 


REFERENCES 


Abramowitz,  M.  and  LA.  Stegun  1964  Handbook  of  Mathematical  Functions,  National 
Bureau  of  Standards,  U.S.  Government  Printing  OfRce,  Washington  D.C.  Reprinted  by 
Dover. 

Bennett,  W.R.,  1944  Acoustical  Society  of  America  15,  165.  Response  of  a  Linear  Rectifier 
to  Signal  and  Noise. 

Bendat,  J.S.,  1958  Principles  and  Applications  of  Random  Noise  Theory,  Wiley,  N.Y. 

Chandrasekhar,  S.,  1943,Reweiys  of  Modem  Physics,  15,  2-74.  Also  available  in  Wax,  N. 
(ed)  Selected  Papers  on  Noise  and  Stochastic  Processes,  Dover,  N.Y.,  1954. 

Cramer,  H.,  1970  Random  Variables  and  Probability  Distributions,  Cambridge  at  the  Uni¬ 
versity  Press. 

Crandall,  S.H.,  and  C.  H.  Mark  1963  Random  Vibrations  in  Mechanical  Systems,  Academic 
Press,  N.Y. 

Department  of  Defense,  1994  Metallic  Materials  and  Elements  for  Aerospace  Vehicle  Struc¬ 
tures,  MIL-HDBK-5G. 

Gradshteyn,  I.S.,  I.M.  Ryzhik,  and  A.  Jeffrey  1994  Table  of  Integrals,  Series,  and  Products 
5th  Ed.,  Academic  Press,  Boston. 

Lin,  P.K.,  1976  Probabilistic  Theory  of  Structural  Dynamics,  Krieger,  reprint  of  1967  edi¬ 
tion  with  corrections. 

Mathematica,  1995  Ver  2.2,  Wolfram  Research,  Champaign,  Illiinois. 

Miles,  J.,  1954  Journal  of  Aeronautical  Sciences  21,  753-762.  On  Structural  Fatigue  under 
Random  Loading. 

Powell,  A.,  1958  Journal  of  the  Acoustical  Society  of  America  SO  No.  12,  1130-1135.  On 
the  Fatigue  Failure  of  Structure  due  to  Vibrations  Excited  by  Random  Pressure  Fields. 

Rayleigh,  J.W.S.  1880  Philosophical  Magazine  X  73-78.  On  the  Resultant  of  a  Large 
Number  of  Vibrations  of  the  Same  Pitch  and  Arbitrary  Phase.  Also  see  Theory  of  Sound, 
Vol  10,  art.  42a,  reprinted  1945  by  Dover,  N.Y..  and  Scientific  Papers,  Dover,  N.Y.,  1964, 
Vol.  I,  pp.  491-496. 

Rice,  S.O.,  1944  The  Bell  System  Technical  Journal  23  282-332.  Continued  in  1945  24  , 
46-156.  Mathematical  Analysis  of  Random  Noise.  Also  available  in  Wax,  N.  (ed)  Selected 
Papers  on  Noise  and  Stochastic  Processes,  Dover,  N.Y.,  1954. 

Shmulei,  U.  and  G.H.  Weiss  1990  Journal  of  the  American  Statistical  Association  85  6-19. 
Probabilistic  Methods  in  Crystal  Structure  Analysis. 

Sveshnikov,  A. A,  1968  Problems  in  Probability  Theory,  Mathematical  Statistics  and  Theory 
of  Random  Functions  Dover,  N.Y.,  translation  of  1965  edition,  pp.  74,  116. 


891 


Tolstov,  G.P.,  1962  Fourier  Series,  Dover,  N.Y.,  pp.  173-177.  Reprint  of  1962  edition. 

Weiss,  G.H.,  1994  Aspects  and  Applications  of  the  Random  Walk,  North-Holland,  Amster¬ 
dam. 

Weiss,  S.H.  and  U.  Shmulei,  1987  Physica  146A  641-649.  Joint  Densities  for  Random 
Walks  in  the  Plane. 

Willie,  L.T.,  1987  Physica  141 A  509-523.  Joint  Distribution  Function  for  position  and 
Rotation  angle  in  Plane  Random  Walks. 

Wirsching,  RH.,  T.L.  Paez,  and  K.  Ortiz  1995  Random  Vibrations,  Theory  and  Practice, 
Wiley-Interscience,  N.Y.,  pp.  162-166. 


892 


A 

an 

Bn 

C{f) 

cih.h) 

E[N,] 

E[No^] 

iFi 

pEq 

f 

i 

3 

Jo 

k 

K 

Ly 

Ey 

m 

N 

Nf 

n 

Bviy) 

Py(^) 

pxYi^,y) 

S 

t 

T 

Y 

Y 
X 

aij 

r 

lij 

(f>n 


U 

a>n 


NOMENCLATURE 

amplitude,  peak,  or  envelope 

amplitude  of  the  nth  sine  wave,  a^,  >  0 

fitted  parameter  in  equation  (42) 

characteristic  function  with  parameter  / 

joint  characteristic  function  with  parameters  fi  and  /2 

expected  number  of  positive  peaks  per  unit  time 

expected  number  of  zero  crossing  with  positive  slope  per  unit  time 

confluent  hypergeometric  function  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  art.  9.210) 

generalized  hypergeometric  function  (Gradshteyn,  Ryzhik,  Jeffrey,  1994,  art. 

9.210) 

parameter  in  Fourier  transform 
integer  index 
imaginary  constant, 

Bessel  function  of  first  kind  and  zero  order 
integer  index 

complete  elliptic  integral  of  first  kind,  equation  (33a) 

+  ^2  +  ••  +  sum  of  amplitudes 
ujiai  +  uj2a2  +  -  •  +  sum  of  velocity  amplitudes 

integer  index 
number  of  terms  in  series 
cycles  to  failure 
integer  index,  n=l,2,..N 

cumulative  probability,  the  integral  of  Py{x)  from  x=— co  to  y 
probability  density  of  random  parameter  Y  evaluated  at  T  =  a: 
joint  probability  density  of  X  and  Y  evaluated  at  Y  =  y  and  X  =  x 
stress 

time,  0  <  t  <  T 

length  of  time  interval 

sum  of  N  modes  or  terms,  —Ly<Y  <  Ly 

first  derivative  with  respect  to  time  of  Y,  —Ly  <Y<  Ly 

a  random  variable 

dimensionless  coefficient,  equation  (33) 

gamma  function,  r[(2n  +  l)/2]  =  7r^/^2“’^(2n  -  1)!! 

dimensionless  coefficient,  equation  (36) 

Dirac  delta  function 
3.1415926.. 

XiX2.-xi\i,  product  of  terms 

phase  angle  of  the  nth  sine  wave,  a  uniformly  distributed  independent  random 
variable 

circular  frequency,  a  positive  (non  zero)  real  number 

circular  frequency  of  the  nth  term,  a  non  zero  integer  multiple  of  27r/T 


893 


Figure  1  Spectrum  of  vibration  of  a  component  on  a  turbojet  engine  cowling.  Note  the 
finite  number  of  distinct  peaks. 


894 


Figure  2  Sample  of  the  time  history  associated  with  the  spectrum  of  Figure  1.  Note  the 
signal  is  bounded,  irregular  and  quasi  sinusoidal. 


895 


-4.0  0  0.0  2.0 

MO,  OF  S.D. 


Figure  3  Probability  density  of  the  time  history  of  Figures  1  and  2.  Note  that  the  maximum 
values  do  not  exceed  plus  or  minus  2.5  standard  deviations. 


Yrms  *  (Probability  Density  of  Y) 


-3-2-10  1  2  3 

Y  /  Yrms 


- Normal  Distribution 

- Sine  Wave  Distribution 

o  Equation  .  N=10 
•  Equation  ,  N=1 


Figure  5  Normal  probability  density  (equation  18)  and  sine  wave  probability  density  (equa¬ 
tion  6)  in  comparison  with  results  of  numerical  integration  of  equation  (13)  for  N=1  and 
N=10. 


898 


Yrms*  (Probability  Density  of  Y) 


Yrms  (Probability  Density  of  A) 


- Two  Sine  Waves 

- Three  Sine  Waves 

. Four  Sine  Waves 


Figure  7  Probability  density  of  peaks  in  narrow  band  series  with  equal  amplitudes  (ai  = 
02..  =  1)  and  frequencies. 


900 


10* 


in'  {o'  10*  10*  10* 

FRTIGUE  LIFE.  CYCLES 

FIGURE  3.2.3.1.8(h).  Besi-fit  SI  N  curves  for  noiched.  K,  ~  4.0  of  2024-T3  aluminum  alloy  sheet, 
longitudinal  direction. 


Figure  8  Fatigue  curves  for  notched  2024-T3  aluminum  alloy  with  Kt=4.  MI1-HDBK-5G 
(1994,  p.  3-115) 


902 


ACOUSTIC  FATIGUE  I 


Strain  Power  Spectra  of  a  Thermally  Buckled  Plate 
in  Random  Vibration 

Jon  Lee  and  Ken  R.  Wentz 
Wright  Laboratory  (FIB) 

Wright-Patterson  AFB,  OH  45433,  USA 

Abstract 

Several  years  ago,  Ng  and  Wentz  reported  strain  power  spectra  measured 
at  the  mid-point  of  a  buckled  aluminum  plate  which  is  randomly  excited  by  an 
electrodynamic  shaker  attached  to  the  clamped-plate  boundary  fixture.  We 
attempt  to  explain  the  peculiar  features  in  strain  power  spectra  by  generating 
the  corresponding  power  spectra  by  the  numerical  simulation  of  a  single-mode 
equation  of  motion.  This  is  possible  because  the  essential  dynamics  takes  place 
in  the  frequency  range  just  around  and  below  the  primary  resonance  frequency. 

1.  Introduction 

For  high  performance  military  aircraft  and  future  high-speed  civil  transport 
planes,  certain  structural  skin  components  are  subjected  to  very  large  acoustic 
loads  in  an  elevated  thermal  environment  [1].  This  is  because  high-speed 
flights  call  for  a  very  powerful  propulsion  system  and  thereby  engendering 
acoustic  loads  in  the  anticipated  range  of  135-175  dB.  More  importantly, 
because  of  the  aerodynamic  heating  in  hypersonic  flights  and  the  modern  trend 
in  integrating  propulsion  sub-systems  into  the  overall  vehicular  configuration, 
some  structural  components  must  operate  at  high  temperatures  reaching  up  to 
1300°F.  Hence,  the  dual  effect  of  thermal  and  acoustic  loading  has  given  rise 
to  the  so-called  thermal-acoustic  structural  fatigue  [2,3]. 

Generally,  raising  the  plate  temperature  uniformly  but  with  an  immovable 
edge  boundary  constraint  would  result  in  thermal  buckling,  just  as  one  observes 
flexural  buckling  as  the  inplane  stress  along  plate  edges  is  increased  beyond  a 
certain  critical  value.  This  equivalence  has  been  recognized  [4,5]  and 
exploited  in  previous  analytical  and  experimental  investigations  of  the  thermal- 
acoustic  structural  fatigue  [6,7,8].  An  experimental  facility  for  thermal- 
acoustic  fatigue,  termed  the  Thermal  Acoustic  Fatigue  Apparatus,  was 
constructed  at  the  NASA  Langley  Research  Center  in  the  late  80’ s.  Under  the 
acoustic  loading  of  140-160  dB,  Ng  and  Clevenson  [9]  obtained  some  strain 
measurements  of  root-mean-square  value  and  power  spectral  density  (PSD)  on 
an  aluminum  plate  heated  up  to  250^.  Later,  Ng  and  Wentz  [10]  have 
repeated  the  heated  Aluminum  plate  experiment  but  by  randomly  exciting  the 
clamped-plate  boundary  fixture  by  a  shaker,  and  thereby  recovering  similar 
strain  measurements. 

It  should  be  noted  that  Ng  and  his  colleagues  [7,9,10]  were  the  first  to 
achieve  sufficient  plate  heating  to  induce  thermal  buckling  and  thus  observe  the 
erratic  snap-through  under  the  acoustic  or  shaker  excitations.  Here,  by  erratic 


903 


we  mean  that  a  snap-through  from  one  static  buckled  position  to  another  takes 
place  in  an  unpredictable  fashion.  We  reserve  the  adjective  chaotic  for  a  snap- 
through  occurring  under  the  deterministic  single-frequency  forcing  [1 1,12].  It 
has  already  been  observed  that  certain  of  the  buckled  plate  experiment  can  be 
explained,  at  least  qualitatively,  by  a  single-mode  model  of  plate  equations. 
This  is  also  validated  by  a  theoretical  analysis.  Indeed,  we  showed  that  a  single¬ 
mode  Fokker-Planck  formulation  can  predict  the  high-temperature  moment 
behavior  and  displacement  and  strain  histograms  of  thermally  buckled  plates, 
metallic  and  composite  [13,14]. 

In  retrospect,  a  single-mode  model  has  proven  more  useful  than  originally 
intended.  That  is,  the  single-mode  Fokker-Planck  formulation  of  an  isotropic 
plate  lends  itself  to  predicting  certain  statistics  of  composite  plates  which  are 
simulated  by  multimode  equations  or  tested  experimentally  by  multimode 
excitations.  For  a  refined  and  more  quantitative  comparison,  one  must  inject 
more  realism  into  dynamical  models  by  including  the  multimode  interactions. 
However,  before  giving  up  the  single-mode  plate  equation,  there  is  an 
important  problem  that  this  simple  model  is  well  suited  for  investigation.  That 
is,  prediction  of  the  strain  PSD  measurement  by  Ng  and  Wentz  [10].  As  we 
shall  see  in  Sec.  4,  the  strain  PSD  of  a  thermally  buckled  plate  exhibits  a  strong 
spectral  energy  transfer  toward  zero  frequency,  and  thereby  saturating 
frequency  range  well  below  the  primary  resonance  frequency.  This  downward 
spectral  energy  transfer  can  be  modeled  quite  adequately  by  the  single-mode 
plate  equation  without  necessitating  multimode  interactions. 

2.  Equation  of  motion  for  the  aluminum  plate  experiment 

By  the  Galerkin  procedure,  the  von  Karman-Chu-Herrmann  type  of  large- 
deflection  plate  equations  give  rise  to  infinitely  coupled  modal  equations  [15]. 
However,  much  has  been  learned  from  a  prototype  single-mode  equation  for 
displacement^  [13,14]. 

q  +  Pq  +  k„{l-s)q  +  aq^  =  g„  +  g{t),  (1) 

where  the  overhead  dot  denotes  d/dt  and  the  viscous  damping  coefficient  is 
P  =  2^^  with  damping  ratio  ^ .  For  the  clamped  plate,  we  have 

;i„=f(r‘'+2rV3  +  l), 

s = rji + (1  -M)  (1 + (r^+  ir^)  /6] , 
a  =  ^{(7^+r'^+2^i)  +  |(i-/i^)[T(r^+r'^)  +^(.r+r''T^ 

+  (r+47'‘)‘^  +  (47+  7"‘r^]} , 

&=  (r‘'+2r^/3  +i)Sjj6. 

Note  that  the  expressions  for  s  and  g„  are  specific  to  the  typical  temperature 


904 


variation  and  gradient  profiles  assumed  in  Ref.  [15].  Here,  7  =  b/a  is  the 
aspect  ratio  of  plate  sides  a  and  b,  and  fi  is  Poisson's  ratio.  The  uniform  plate 
temperature  is  measured  in  units  of  the  critical  buckling  temperature.  The 
maximum  temperature  variation  on  the  mid-plate  plane  is  denoted  by  and 
TJ5g  is  the  maximum  magnitude  of  temperature  gradient  across  the  plate 
thickness,  where  5^  and  5 ^  are  scale  factors.  Hence,  0  signifies  no 
temperature  variation  over  the  mid-plate  plane,  and  0  zero  temperature 
gradient  across  the  plate  thickness.  Finally,  g{t)  denotes  the  external  forcing. 

The  parameter  s  represents  thermal  expansion  due  to  both  the  uniform 
plate  temperature  rise  above  room  temperature  and  temperature  variation  over 
the  mid-plate  uniform  temperature.  The  combined  stiffness  k^(l  -  s)q  consists 
of  the  structural  stiffness  k^q  and  thermal  stiffness  -sk^q ,  which  cancel  each 
other  due  to  the  sign  difference.  It  is  positive  for  5  <1,  then  Eq.  (1)  has  the 
form  of  Duffing  oscillator  with  a  cubic  term  multiplied  by  a ,  which  represents 
geometric  nonlinearity  of  membrane  stretching.  For  s  >1  Eq.  (1)  reduces  to 
the  so-called  buckled-beam  equation  of  Holmes  [11]  with  a  negative  combined 
stiffness.  In  contrast,  denotes  thermal  moment  induced  by  a  temperature 
gradient  across  the  plate  thickness;  hence,  it  appears  in  the  right-hand  side  of 
Eq.  (1)  as  an  additional  forcing.  The  interplay  of  the  terms  involving  5,  a,  and 
g^  can  best  be  illustrated  by  the  potential  energy  [15] 

U{q)  =  -go q  +  k^(X-s) (fn  -H  a .  (2) 

Fig.  1  shows  that  V{c^  is  symmetric  when  g„  =  0.  For  s<l  it  has  a  single  well 
which  splits  into  a  double  well  as  s  exceeds  unity.  Note  that  the  distance 
between  the  twin  wells  increases  as  for  large  s  (Fig.  1(b)).  This 
interpretation  is  valid  approximately  for  go>^-  That  is,  a  positive  g^  lowers 
the  positive  side  potential  (^>0)  and  raises  the  negative  side  potential  {q<0), 
and  thereby  rendering  the  potential  energy  asymmetric. 

U(q)  ^(^1) 


Fig.  1  Potential  energy,  (a)  s<V,  (b)  ^  >1,  w  =  ^k^(s  -  l)/a  ,  d  =  -  l))V4a. 

( -  ^.=0;  ---  5„>0) 


905 


It  must  be  pointed  out  that  Eq.  (1)  is  dimensionless  and  involves  explicitly 
only  7  and  /i.  For  the  aluminum  plate  experiment  [10],  7=10  in./8  in.  and 
so  that  k^=Q3.9l  and  a =85. 33.  If  we  further  assume  5^-  0  for 

simplicity,  the  thermal  parameter  reduces  to  s=T^.  Previously,  Eq.  (1)  was 
used  for  the  investigation  of  stationary  Fokker-Planck  distribution  which 
involves  only  the  ratio  p!F,  where  F  is  the  constant  power  input  [13,14]. 
Hence,  nondimensionalization  has  indeed  spared  us  from  specifying  in  detail 
other  plate  parameters.  Things  are  however  different  in  numerical  simulation 
because  we  must  know  the  characteristic  scales  to  correctly  interpret  time- 
dependent  solutions.  By  retracing  the  derivation,  we  find  that  the  dimensionless 
quantities  in  Eq.  (1)  are  (Eq.  (IV.  1)  in  Ref.  [1]) 

q-qlh,  t-t/t*,  g  =  glg*,  (3) 

where  the  overhead  bar  denotes  the  physical  quantity.  Here,  the  plate  thickness 
h,  t*={b/Kf.y[ph/D,  and  g^=p(h/t*)^  are  the  characteristic  length,  time,  and 
force,  respectively  (p  =  mass  density,  D=Eh^f\2{\~p}) ,  £=  Young’s 
modulus  of  elasticity),  as  listed  in  Table  L  We  now  rewrite  Eq.  (1)  with  the 
numerical  coefficients  (Table  I).  _ 

q  +  0.0978?  +  23.91(1  -s)q  +  85.33?^  =  (4) 


where  g{t)  has  the  unit  of  psi. 

Table  I.  Parameter  values  for  the  aluminum  plate  experiment 

7,  « 

10in./8in.,  VoX  0.01,  23.910,  85.332 

5..  5, 

p 

o 

h,  f*,  g'^- 

0.05  in.,  3.305  lO'^sec.^,  5.806  1  O'’ psi^ 

(+)  p  =  0.098  Ib/in^  and  E  =1.03  10’  psi. 


3.  Monte-Carlo  simulation 

Because  of  5  =0,  Eq.  (4)  has  the  standard  form  of  Duffing  (s<l)  and 
Holmes  (s>l)  oscillators.  In  stead  of  a  single  frequency  for  forcing  g{t)  [11, 
12],  in  Monte-Carlo  simulation  all  forcing  frequencies  are  introduced  up  to  a 
preassigned  maximum  so  that  forcing  represents  a  plausible  physical 
realization.  Of  course,  particular  interest  here  is  a  constant  PSD.  We  shall 
begin  with  generation  of  a  time-series  for  random  processes  with  such  a  PSD. 

3.1  Random  forcing  time-series 

We  adopt  here  the  procedure  for  generating  a  time-series  of  Shinozuka 
and  Jan  [16],  which  has  been  used  for  a  oscillator  study  [17]  and  extensively 
for  structural  simulation  applications  by  Vaicaitis  [18,19].  Since  it  relies 
heavily  on  the  discrete  fast  Fourier  transforms,  such  as  FFTCF  and  FFTCB 
subroutines  of  the  IMSL  library,  it  is  more  expedient  to  describe  the  procedure 
operationally  rather  than  by  presenting  somewhat  terse  formulas.  Let  us 


906 


introduce  Nj:  frequency  coordinates  which  are  equally  spaced  in 
the  band  width  A/=/^3x/A^^.  Now,  the  task  is  to  generate  a  time-series  of  total 
time  T  that  can  resolve  up  to  .  Assume  T  is  also  divided  into  time 
coordinates  with  the  equal  time  interval  At=T/Nj,  From  the  time-frequency 
relation  r=l/A/,  we  find  ^At.  If  we  choose 

N 

Nf  =  (5) 

is  the  Nyquist  frequency,  consistent  with  our  original  definition  of  the 
upper  frequency  limit  of  resolution. 

A  random  time-series  with  a  constant  PSD  can  be  generated  in  the 
following  roundabout  way.  We  begin  by  assuming  that  we  already  have  a 
forcing  power  spectrum  ^g{f)  of  constant  magnitude  over  [0,/njax]-  Such  a 

PSD  may  be  represented  by  a  complex  array  A„= VC  exp(~27rz0„)  (n  =  1, 
Nf),  where  takes  a  random  value  distributed  uniformly  in  [0,  1].  Clearly  the 
magnitudes  of  are  C,  hence  We  then  enlarge  the  complex  array 

A„  by  padding  with  zeros  for  n  =Ny+l,  and  Fourier  transform  it  to 

obtain  a  complex  array  B^{n=  N^).  The  random  time-series  for  is 

now  given  by  the  real  part  of 

=  Real  part  of  (n  =  1, N^)  (6) 


As  it  turns  out,  when  g  „  is  padded  with  zeros  for  the  imaginary  components 
and  Fourier  transformed,  we  recover  the  original  array  A„  (w=l,  Nf)  with 


Since  the  spectrum  area  is  nothing  but  total  forcing  power 

<g^>  (say,  in  psi^),  we  can  relate  C  with  the  variance  <g^>  of  pressure 
fluctuations,  which  is  often  expressed  by  the  sound  pressure  level  (SPL)  in  dB, 
according  to  SPL=10  log<g2  >/p2,  where  p=2,9  10*^  psi.  Hence, 

c  =  - .  (7) 

/max 

Here,  Eqs.  (6)  and  (7)  defined  heuristically  are  meant  to  explain  the 
corresponding  formulas  (2)  and  (12)  in  Ref  [18]. 

For  the  numerical  simulation  we  first  note  that  the  resonance  frequency  of 
Eq.  (1)  is  /^=-y^/27r«  0.778  for  s  ~  0.  This  gives  the  dimensional  resonance 

frequency  fjt*~235.5  Hz  which  is  somewhat  larger  than  the  experimental 
217.7  Hz  (Fig.  3(a)).  As  shown  in  Table  II,  we  assign  (~9/r)  because 

the  electrodynamic  shaker  used  in  the  experiment  [10]  has  the  upper  frequency 
limit  2000  Hz. 


Table  II.  Dimensionless  parameter  values  for  the  numerical  simulation 

at,  N, _ 7,  8192,  4096 

At,  T  0.071.  585 


907 


3.2  Displacement  power  spectrum 

Under  a  random  time  integration  of  Eq.  (4)  yields  a  time-series  for  q^. 
We  first  comment  on  the  time  integration.  Although  there  are  special  solvers 
[17,20]  proposed  for  stochastic  ordinary  differential  equations  (ODEs),  we 
shall  use  here  the  Adams-Bashforth-Moulton  scheme  of  Shampine  and  Gordon 
[21],  which  has  been  implemented  in  DEABM  subroutine  of  the  SLATEK 
library.  Although  DEABM  has  been  developed  for  nonstochastic  ODEs,  its 
use  for  the  present  stochastic  problem  may  be  justified  in  part  by  that  one 
recovers  linearized  frequency  response  functions  by  the  numerical  simulation 
(Sec.  3.4).  Obviously,  this  does  not  say  anything  about  the  strongly  nonlinear 
problem  in  hand,  and  it  should  be  addressed  as  a  separate  issue.  In  any  event, 
DEABM  requires  the  absolute  and  relative  error  tolerances,  both  of  which  are 
set  at  no  larger  than  10“^  under  the  single-precision  algorithm  for  time 
integration.  Note  that  actual  integration  time  steps  are  chosen  by  the 
subroutine  itself,  commensurate  with  the  error  tolerances  requested.  Recall 
that  is  updated  at  every  time  interval  Ar,  and  we  linearly  interpolate  the 
forcing  value  within  A? . 

We  begin  time  integration  of  Eq.  (4)  from  the  initial  configuration  at  the 
bottom  of  the  single-well  potential,  ^(0)=p(0)=0,  for  5'  <1  and  the  positive 
side  double-well  potential,  q{0)=^kj,s~\)/a  andp(0)=0,  for  j  >1.  And  we 
continue  the  integration  up  to  T.  By  Fourier  transforming  time-series  q^,  we 
obtain  displacement  power  spectrum  0^(/).  This  process  of  integrating  and 
transforming  is  repeated  over  three  contiguous  time  ranges  of  7,  and  the 
successive  PSDs  are  compared  for  stationarity.  Since  it  is  roughly  stationary 
after  three  repetitions,  we  report  here  only  the  PSD  of  the  third  repetition. 
From  the  stationary  input-output  relation  [22]  where 

is  the  magnitude  of  system  frequency  response  function,  we  write 

=  (S) 

Since  O  (/)=C,  the  and  would  have  a  similar  functional 

s 

dependence  upon  /,  Hence,  we  call  them  both  the  displacement  PSD. 

3.3  Strain  power  spectrum 

Although  displacement  is  the  direct  output  of  numerical  simulation,  one 
measures  strain  rather  than  the  displacement  in  plate  experiment.  At  the 
present  level  of  plate  equation  formulation,  the  strain  e  is  given 

by  the  quadratic  relation 

e  =  +  C^q  +  C2(f‘  ■>  (9) 

where  C,-  are  given  at  the  middle  {x/a  =  y/b  =1/2)  of  a  clamped  plate  as  follows 
(Appendix  D  of  Ref  [13]) 


908 


c  = 


(/+2yV3+l)?;g, 

3(1+At)(y"+1) 


{l-li}- 


q  = 


8y^ 

3  ’ 


r  32  fy^  5n  (l-/iyV4) 

^  9  [2  16  2(y+y-‘f  (y+4y-‘)^  (4y  +  y‘fJ 


For  we  have  C^=0,  Q  =4.17,  and  Q  =2.77  (Table  I).  Hence,  Eq.  (9) 
engenders  only  the  linear  and  quadratic  transformations,  but  no  translation.  In 
any  event,  translation  has  no  effect  on  the  spectral  energy  contents.  By  Fourier 
transforming  time-series  (n=l,  A^^),  we  obtain  strain  power  spectrum 

.  Although  the  forcing  PSD  is  not  constant,  one  computes  the  forcing 
spectrum  ratio  as  in  Eq.  (8)  and  call  it  the  magnitude  square  of  strain  frequency 
response  function  for  the  lack  of  a  better  terminology. 

3.4  The  linear  oscillators 

For  the  pre-buckled  (5  <1)  linear  oscillator  (a=  0)  we  rewrite  Eq.  (4)  in 
standard  form  _ 

q  +  +  0)1(1  -s)q  =  (10) 

where  col=k^,  and  obtain 

I  H^(f)^  =  [(0)1(1  -s)-  +  (An^co^ffT'-  (11) 

As  shown  in  Fig.  2(a),  the  numerical  simulation  of  Eq.  (10)  recovers 
as  given  by  Eq.  (11)  over  the  entire  frequency  range.  Although  the  simulation 
of  Fig.  2(a)  was  carried  out  with  SPL=130  dB,  it  does  not  depend  on  SPL 
since  Eq.  (10)  is  linear.  Physically  speaking,  Eq.  (10)  oscillates  in  a  single-well 
potential  (Fig.  1(a)).  Since  the  potential  energy  has  two  wells  (Fig.  1(b))  for 
s  >1,  we  linearize  Eq.  (1)  around  the  positive  side  potential  well  by  the 
transformation  q=q'+^k^{s-l)la .  Hence,  the  corresponding  linear  oscillator 

is 


(a)  (b) 


Fig.  2  Linear  frequency  response  functions,  (a)  Displacement;  (b)  Strain 
(j  =  0;  - Numerical  simulation;  •  Eq.  (U)) 


909 


q'  +  2^0)^  q’  +  Icolis  -V)q'  =  (12) 

s 

In  parallel  to  Eq.  (1 1),  the  frequency  response  function  of  a  post-buckled  plate 

I  =  [(^-colis  -  1)  -  4;rV^)2  +  .  (13) 

The  resonance  frequency  f=co^^2{s-l)/27J:  of  a  post-buckled  (s  >1)  plate 
should  be  compared  with  f=co^^2{\~s)/2n  of  the  pre-buckled  (j  <1)  plate. 

Now,  for  the  linear  oscillators  we  see  that  is  also  given  by  Eq.  (1 1) 

and  (13)  for  ^  <1  and  >1,  respectively  (Fig.  2(b)).  This  is  because  the  spectral 
energy  distribution  is  not  at  ail  affected  by  a  linear  transformation. 

4.  Displacement  and  strain  power  spectra 

As  we  shall  see  in  Sec  4.1,  the  experimental  strain  PSD  exhibits  downward 
spectral  energy  transfer  toward  zero  frequency,  so  that  there  is  a  considerable 
spectral  energy  buildup  below  the  resonance  frequence  as  SPL  is  raised. 
Moreover,  it  also  involves  an  upward  spectral  energy  transfer  which  then 
contributes  to  both  the  increased  resonance  frequency  and  broadened 
resonance  frequency  peak.  Since  spectral  energy  transfers  take  place  around 
and  below  the  primary  resonance  frequency,  it  is  possible  to  depict  the 
downward  and  upward  spectral  energy  transfers  by  the  numerical  simulation  of 
Eq.  (4)  without  necessitating  multimode  interactions.  We  shall  first  discuss  the 
characteristic  features  of  experimental  strain  PSDs. 

4. 1  Experimental  strain  PSD 

Of  the  spectra  reported  in  Ref.  [10],  we  consider  the  following  two  sets. 
One  is  the  nonthermal  set  (^=0)  consisting  of  two  PSDs  of  small  and  large 
SPLs.  The  other  is  the  post-buckled  set  (5=1.7)  of  four  PSDs.  For  the 
convenience  of  readers,  we  have  reproduced  in  Figs.  3  and  4  the  selected  PSDs 
from  Ref.  [10]  by  limiting  the  upper  frequency  to  600  Hz,  and  the  pertinent 
data  are  summarized  in  Table  m. 


Table  in.  Strain  power  spectra  of  experiment  and  numerical  simulation 


Fig.  4(a)  Fig.  6a 


Fig.  4(b)  Fig.  6b 


Fig.  4(c)  Fig.  6c 
Fig.  4(d)  Fig.  6d (*) 


Fig.  7 
Fig.  8 
Fig.  9 


Fig.  10 


(*)  Computed  from  the  acceleration  a  measured  in  units  of  g. 


910 


Fig.  4  Experimental  strain  PSD  =1.7).  (a)  130.1dB;  (b)  142dB;  (c)  151.5dB;  (d)  154.6dB 

The  following  observations  are  drawn  from  the  experimental  PSDs.  First, 
for  the  nonthermal  plate 

Figure  3 fa):  Compare  the  measured  strain  fr-2\l  Hz  with  the  theoretical 
displacement 235  Hz  of  Eq.  (4).  Note  that  a  small  spectral  energy  peak  is 
found  at  467  Hz  which  is  about  twice  (-2.15)  the  strain  value. 


911 


Figure  Sfb):  With  SPL~150  dB  the  strain  increases  to  240  Hz  and  the 
spectral  width  at  the  half  resonance  peak  has  nearly  doubled.  The  spectral 
energy  buildups  at  zero  and  515  Hz  are  more  noticeable  than  in  Fig.  3(a). 
Again,  515  Hz  is  about  twice  (-2.15)  the  primary  strain  f,. 


(a)  (b) 


f  f 


Fig.  5  Numerical  simulation  results  under  .y=0  and  SPL=130  dB. 

(a)  Displacement  ( - simulation,*  Eq.  (11));  (b)  Strain  ( - simulation,  •  Eq.  (11)); 

(c)  PSD  averaged  over  12  frequency  intervals  ( - displacement,  — •  —  strain); 

(d)  Strain  PSD. 


Next,  for  the  thermally  buckled  plate 

Figure  4(a):  The  primary  strain  fr=227  Hz  should  be  compared  with  the 
theoretical  displacement  /^=279  Hz  of  Eq.  (13).  A  second  spectral  energy 
peak  is  found  at  537  Hz,  much  larger  than  twice  (-2.37)  the  primary  strain  /^. 
Figure  4rb):  Here,  the  spectral  energy  buildup  is  most  significant  at  zero 
frequency.  Besides,  there  appear  two  spectral  energy  humps  at  100  and  183 
Hz,  below  the  primary  strain  =  227  Hz  of  Fig.  4(a).  Discounting  the  zero- 
frequency  spectral  peak,  PSD  may  be  approximated  by  a  straight  line  in  the 
semi-log  plot,  hence  it  is  of  an  exponential  form  up  to  400  Hz. 


Figure  4rc):  The  zero-frequency  peak  is  followed  by  a  single  spectral  energy 
hump  at  115  Hz.  Again,  PSD  can  be  approximated  by  a  straight  line  and  its 
slope  is  roughly  the  same  as  in  Fig.  4(b). 

Figure  4rd):  A  major  spectral  energy  peak  emerges  at  130  Hz,  followed  by  a 
minor  one  at  350  Hz.  Theoverall  spectral  energy  level  is  raised  so  that  the 
magnitude  of  PSD  ranges  over  only  two  decades  in  the  figure. 

In  Figs.  4(b)-(d)  we  have  ignored  the  spectral  energy  peaks  at  around  500 
Hz,  for  they  are  not  related  to  the  first  plate  mode  under  consideration.  This  is 
further  supported  by  the  simulation  evidence  to  be  discussed  presently. 


4.2  Numerical  simulation  results 

After  choosing  .y  =  0  or  1.7,  we 
are  left  with  SPL  yet  to  be  specified.  o 
Ideally,  one  would  like  to  carry  out  the 
numerical  simulation  of  Eq.  (4)  by 
using  SPL  of  the  plate  experiment  'm-z 
(Table  III)  and  thus  generate  strain  ^ 

PSDs  which  are  in  agreement  with 
Figs.  3  and  4.  Not  surprisingly,  the  _4 
reality  is  less  than  ideal.  An  obvious 
reason  that  this  cannot  be  done  is  that 
the  forcing  energy  input  is  fed  into  all  f 

plate  modes  being  excited  in  Hg.  6  PSD  averaged  over  12  frequency 
experiment,  whereas  the  forcing  (j=0,  SPL=138dB) 

energy  excites  only  one  mode  in  the  - displacement;  -•-strain 

numerical  simulation.  Consequently,  SPL  for  the  numerical  simulation  should 
be  less  than  the  experimental  SPL,  but  we  do  not  know  a  priori  how  much 
less.  We  therefore  choose  a  SPL  to  bring  about  qualitative  agreements 
between  the  single-mode  simulation  and  multimode  experiment.  As  anticipated, 
the  simulation  SPLs  (Table  HI)  are  consistently  smaller  than  the  experimental 
values. 

The  numerical  simulation  results  are  shown  in  Figs.  5-6  for  5  =  0  and  Figs. 
7-10  for  s  =1.7.  Actually  each  figure  has  four  frames,  denoted  by  (a)-(d). 
First,  frames  (a)  and  (b)  depict  and  Since  they  are  very 

jagged  at  large  SPLs,  we  average  the  spectral  energy  over  12  frequency 
intervals  and  present  both  of  the  smoothed-out  frequency  response  functions  in 
the  same  frame  (c).  Lastly,  frame  (d)  shows  Og(/)  itself  Since  there  is  no 
qualitative  difference  between  <E>g(/)  and  we  shall  call  them  both  the 

strain  PSD.  We  present  all  four  frames  (a)— (d)  of  Figs.  5  and  7,  but  only  the 
frame  (c)  of  Figs.  6,  8,  9  and  10  here  for  the  lack  of  space. 

First,  for  the  nonthermal  plate 

Figure  5:  The  simulated  is  closely  approximated  by  Eq.  (11)  with  f  = 

236  Hz.  Note  that  is  also  approximated  by  Eq.  (11)  for  all  frequencies 


913 


Fig.  7  Numerical  simulation  results  under  j=1.7  and  SPL=129  dB. 

(a)  Displacement  ( - simulation,*  Eq.  (13));  (b)  Strain  ( - simulation,  •  Eq.  (13)); 

(c)  PSD  averaged  over  12  frequency  intervals  ( - displacement,  — •  strain); 

(d)  Strain  PSD. 


but  zero  and  476  Hz,  where  the  strain  spectral  energy  piles  up  due  to  the 
quadratic  transformation  (9).  Since  476  Hz  is  nearly  twice  (-2.02)  the  primary 
/^,  strain  spectral  energy  buildups  are  due  to  the  sum  and  difference  of  two 

nearly  equal  frequencies,  ±  /2,  where/i==/2^/^. 

Figure  6:  The  primary  strain  is  shifted  slightly  upward  to  253  Hz  and  the 

spectral  width  at  half  resonance  peak  is  50%  wider  than  that  of  Fig.  5(c).  The 
spectral  energy  builds  up  at  525  Hz  which  is  roughly  twice  (-2.08)  the  /^.  At 
SPL=138  dB  we  find  that  the  strain  spectral  energy  hump  at  525  Hz  is  about  2 
decades  below  the  resonance  frequency  peak,  as  was  in  Fig.  3(b). 

Now,  for  the  thermally  buckled  plate 

Figure  7:  The  simulated  and  are  weU  approximated  by  Eq. 

(13)  around  /^=270  Hz  which  is  a  litde  below  the  linearized  /^=279  Hz. 
Unlike  in  Fig.  5  for  5=0,  both  and  l/7^(/)F  show  spectral  energy 

building  up  significantly  near  zero  and  543  Hz  which  is  twice  (-2.01)  the  /^. 


914 


2p — ! - 1 - ! - 1 - 1 - T— n - 1 - r 


^ _ 1 - 1 - 1 - < - 1 - > - 1 - ^ - 1 

0  300  600 


f 

Fig.  8  PSD  averaged  over  12  frequency 
intervals  (j  =1.7,  SPL=138  dB) 

- displacement;  — •  —  strain 


Note  that  in  Fig.  7(a)  the  spectral 
energy  hump  at  543  Hz  is  about  3 
decades  below  the  primary  frequency 
peak,  as  was  in  Fig.  4(a). 

Figure  8:  After  a  large  zero-frequency 
peak,  two  spectral  energy  humps 
appear  at  131  Hz  and  236  Hz.  Note 
that  the  ratios  of  these  frequencies  to 
the  /,  (131/279  -0.47  and  236/279  - 
0.85)  are  comparable  with  the  same 
ratios  (100/227  -0.44  and  183/227  - 
0.81)  found  in  Fig.  4(b).  Excluding 
the  zero-frequency  peak,  the  overall 
strain  PSD  is  a  straight  line,  hence  of 
an  exponential  form,  as  in  Fig.  4(b), 


Figure  9:  The  zero-frequency  spectral  peak  is  followed  by  a  single  major 
energy  hump  at  154  Hz.  The  ratio  of  this  to  the  (154/279  -0.56)  is 
somewhat  larger  than  the  ratio  (115/227  -0.51)  in  Fig.  4(c).  The  strain  PSD 
can  also  be  approximated  by  a  straight  line  over  the  entire  frequency  range,  and 
Figs.  8  and  9  seem  to  have  the  same  slope  when  fitted  by  straight  lines. 

Figure  10:  The  spectral  magnitude  of  is  larger  than  that  of  in 

the  frequency  range  above  300  Hz.  The  choice  of  SPL=146  dB  was  based  on 
that  the  PSD  magnitude  around  300  Hz  is  about  2  decades  below  the  main 
spectral  peak  magnitude  at  180  Hz,  thus  emulating  Fig.  4(d). 

All  in  all,  by  numerical  simulations  we  have  successfully  reproduced  the 
peculiar  features  in  the  two  sets  of  strain  PSDs  observed  experimentally  under  5 
=  0  and  1.7. 


Fig.  9  PSD  averaged  over  12  frequency 
intervals  (s  =1.7,  SPL=143  dB) 

- displacement;  — • — strain 


Fig.  10  PSD  averaged  over  12  frequency 
intervals  (j  =1.7,  SPL=146  dB) 

- displacement;  — • — strain 


915 


5.  Concluding  remarks 

At  low  SPL  the  nonthermal  {s=  0)  and  post-buckled  (^=1.7)  plates  appear 
to  have  a  similar  PSD.  However,  this  appearance  is  quite  deceptive  in  that  the 
nonthermal  plate  motion  is  in  a  single-well  potential,  so  that  PSD  does  not 
change  qualitatively  as  SPL  is  raised.  On  the  other  hand,  the  trajectory  of  a 
post-buckled  plate  is  in  one  of  the  two  potential  energy  wells  when  SPL  is  very 
small.  However,  as  we  raise  SPL  such  a  plate  motion  can  no  longer  be 
contained  in  a  potential  well,  and  hence  it  encircles  either  one  or  both  of  the 
potential  wells  in  an  erratic  manner.  This  is  why  the  experimentally  observed 
and  numerically  simulated  strain  PSDs  of  a  post-buckled  plate  exhibit 
qualitative  changes  with  the  increasing  SPL,  and  thereby  reflect  the  erratic 
snap-through  plate  motion.  A  quantitative  analysis  of  snap-through  dynamics 
will  be  presented  elsewhere. 

Lastly,  we  wish  to  point  out  that  a  PSD  of  straigh-line  form  in  the  semi-log 
plot  was  observed  in  a  Holmes  oscillator  when  trajectories  are  superposed 
randomly  near  the  figure-eight  separatrix  [23]. 

Acknowledgments 

Correspondence  and  conversations  with  Chung  Fi  Ng,  Chuh  Mei,  Rimas 
Vaicaitis,  and  Jay  Robinson  are  sincerely  appreciated.  We  also  wish  to  thank 
the  referees  for  their  helpful  suggestions  to  improve  the  readability  of  this 
paper. 


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6.  Seide,  P.  and  Adami,  C.,  Dynamic  stability  of  beams  in  a  combined 
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7.  Ng,  C.F.,  Nonlinear  and  snap-through  responses  of  curved  panels  to 
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8.  Robinson,  J.H.  and  Brown,  S.A.,  Chaotic  structural  acoustic  response  of  a 
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Material  Conference,  AIAA-95-1301-CP,  New  Orleans,  LA,  1240-1250, 
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ENHANCED  CAPABELITIES  OF  THE  NASA  LANGLEY 
THERMAL  ACOUSTIC  FATIGUE  APPARATUS 

Stephen  A.  Rizzi  and  Travis  L.  Turner 
Structural  Acoustics  Branch 
NASA  Langley  Research  Center 
Hampton,  VA  23681-0001 

ABSTRACT 

This  paper  presents  newly  enhanced  acoustic  capabilities  of  the  Thermal 
Acoustic  Fatigue  Apparatus  at  the  NASA  Langley  Research  Center.  The 
facility  is  a  progressive  wave  tube  used  for  sonic  fatigue  testing  of  aerospace 
structures.  Acoustic  measurements  for  each  of  the  six  facility  configurations 
are  shown  and  comparisons  with  projected  performance  are  made. 

INTRODUCTION 

The  design  of  supersonic  and  hypersonic  vehicle  stmctures  presents  a 
significant  challenge  to  the  airframe  analyst  because  of  the  wide  variety  and 
severity  of  environmental  conditions.  One  of  the  more  demanding  of  these  is 
the  high  intensity  noise  produced  by  the  propulsion  system  and  turbulent 
boundary  layer  [1].  Complicating  effects  include  aero-thermal  loads  due  to 
boundary  layer  and  local  shock  interactions,  static  mechanical  preloads,  and 
panel  flutter.  Because  of  the  difficulty  in  accurately  predicting  the  dynamic 
response  and  fatigue  of  structures  subject  to  these  conditions,  experimental 
testing  is  often  the  only  means  of  design  validation.  One  of  the  more  common 
means  of  simulating  the  thermal- vibro-acoustic  environment  is  through  the  use 
of  a  progressive  wave  tube.  The  progressive  wave  tube  facility  at  NASA 
Langley  Research  Center,  known  as  the  Thermal  Acoustic  Fatigue  Apparatus 
(TAFA),  has  been  used  in  the  past  to  support  development  of  the  thermal 
protection  system  for  the  Space  Shuttle  and  National  Aerospace  Plane  [2].  It 
is  presently  being  used  for  sonic  fatigue  studies  of  the  wing  strake 
subcomponents  on  the  High  Speed  Civil  Transport  [3]. 

The  capabilities  of  the  TAFA  were  previously  documented  by  Clevenson  and 
Daniels  [4].  The  system  was  driven  by  two  Wyle  WAS  3000  airstream 
modulators  which  provided  an  overall  sound  pressure  level  range  of  between 
125  and  165  dB  and  a  useful  frequency  range  of  50-200  Hz.  A  360  kW  quartz 
lamp  bank  provided  radiant  heat  with  a  peak  heat  flux  of  54  W/cm^.  A 
schematic  of  the  facility  is  shown  in  Figure  1.  Representative  spectra  and 
coherence  plots  are  shown  in  Figures  2  and  3.  Since  that  time,  the  facility  has 
undergone  significant  enhancements  designed  to  improve  its  acoustic 
capabilities;  the  heating  capabilities  were  not  changed.  The  objectives  of  the 
enhancements  were  to  increase  the  maximum  overall  sound  pressure  level 
(OASPL)  to  178  dB,  increase  the  frequency  bandwidth  to  500  Hz  and  improve 
the  uniformity  of  the  sound  pressure  field  in  the  test  section.  This  paper 


919 


documents  the  new  capabilities  of  the  TAFA  and  makes  comparisons  with  the 
projected  performance. 


Figure  1:  Schematic  of  the  old  TAFA  facility. 


Figure  2:  Test  section  spectra  of  the  Figure  3:  Test  section  coherence  of  the 
old  TAFA  facility.  old  TAFA  facility. 


FACILITY  DESCRIPTION 

In  order  to  meet  the  design  objectives,  extensive  modifications  were  made  to 
the  sound  generation  system  and  to  the  wave  tube  itself.  A  theoretical  increase 
of  6  dB  OASPL  was  projected  by  designing  the  system  to  utilize  eight  WAS 
3000  air  modulators  compared  to  the  two  used  in  the  previous  system.  A 
further  increase  of  nearly  5  dB  was  expected  by  designing  the  test  section  to 
accommodate  removable  water-cooled  insert  channels  which  reduced  its  cross- 
sectional  area  from  1.9m  x  0.33m  to  0.66m  x  0.33m.  The  frequency  range  was 
increased  through  the  use  of  a  longer  horn  design  with  a  lower  (15  Hz  vs.  27 
Hz  in  the  old  facility)  cut-off  frequency,  use  of  insert  channels  in  the  test 
section  to  shift  the  frequency  of  significant  standing  waves  above  500  Hz,  and 
design  of  facility  sidewall  stmctures  with  resonances  above  1000  Hz.  The 
uniformity  of  the  sound  pressure  field  in  the  test  section  was  improved  through 
several  means.  A  new,  smooth  exponential  horn  was  designed  to  avoid  the 
impedance  mismatches  of  the  old  design.  To  minimize  the  effect  of 
uncorrelated,  broadband  noise  (which  develops  as  a  byproduct  of  the  sound 


920 


generation  system),  a  unique  design  was  adopted  which  allows  for  the  use  of 
either  two-,  four-,  or  eight-modulators.  When  testing  at  the  lower  excitation 
levels  for  example,  a  two-modulator  configuration  might  be  used  to  achieve  a 
lower  background  level  over  that  of  the  four-  or  eight-modulator 
configurations.  In  doing  so,  the  dynamic  range  is  extended.  Lastly,  a  catenoidal 
design  for  the  termination  section  was  used  to  smoothly  expand  from  the  test 
section. 

Schematics  of  the  facility  in  the  three  full  test  section  configurations  are  shown 
in  Figures  4-6.  In  the  two-modulator  configuration,  the  2  x  4  transition  cart 
acts  to  block  all  but  two  of  the  eight  modulators.  The  facility  is  converted 
from  the  two-  to  four-modulator  configuration  by  the  removal  of  the  2  x  4 
transition  cart  and  connection  of  two  additional  modulators.  In  doing  so,  the 
modulator  transition  cart  slides  forward  and  thereby  maintains  the  continuous 
exponential  expansion  of  the  duct.  In  the  four-modulator  configuration,  the  4 
X  8  transition  cart  acts  to  block  the  two  upper  and  two  lower  modulators. 
Removal  of  this  component  and  connection  of  the  four  additional  modulators 
converts  the  facility  to  the  eight-modulator  configuration.  Again,  the 
continuous  exponential  expansion  is  maintained  as  the  modulator  transition 
cart  slides  forward. 


Figure  4:  Two-modulator  full  test  section  configuration. 


921 


Figure  5:  Four-modulator  full  test  section  configuration. 


Figure  6:  Eight-modulator  full  test  section  configuration. 


Schematics  of  the  three  reduced  test  section  configurations  are  shown  in 
Figures  7-9.  In  these  configurations,  the  horn  cart  is  discarded  and  the  horn 
transition  cart  mates  directly  to  the  test  section.  Water-cooled  inserts  are  used 
in  the  test  section  to  reduce  its  cross-sectional  area.  Upper  and  lower  inserts  in 
the  termination  section  are  used  to  smoothly  transition  the  duct  area  to  the  full 
dimension  at  the  exit.  Conversion  from  the  two-  to  the  four-modular 
configuration  and  from  the  four-  to  the  eight-modulator  configuration  is  again 
accomplished  through  removal  of  the  2  x  4  and  4x8  transition  carts, 
respectively. 


922 


HORN  TTUNSTTION  SECTION 


Figure  7:  Two-modulator  reduced  test  section  configuration. 


Figure  8:  Four-modulator  reduced  test  section  configuration. 

TEST  PROCEDURE 


Measurements  were  taken  for  several  conditions  in  each  of  the  six  facility 
configurations.  Each  modulator  was  supplied  with  air  at  a  pressure  of  207  kPa 
(mass  flow  rate  of  approximately  8.4  kg/s)  and  was  electrically  driven  with  the 
same  broadband  (40-500  Hz)  signal.  Acoustic  pressures  were  measured  at 
several  locations  along  the  length  of  the  progressive  wave  tube  using  B&K 
model  4136  microphones  and  Kulite  model  MIC-190-HT  pressure  transducers, 
see  Table  1.  The  positive  x-direction  is  defined  in  the  two-modulator  full 
configuration  (from  the  modulator  exit)  along  the  direction  of  the  duct.  The 
positive  y-direction  is  taken  vertically  from  the  horizontal  centerline  of  the 


923 


HORN  TRIWSmOH  SECTION 


MODULATOR  TRANSITION 
FLEMSie  HOSE 


adapter  puts  assembly 

TEST  SECTION 

ADAPTER  plate  assembly 
TERM1ASAT10N  SECTION 


horn  TRANSITION  CART 


Figure  9:  Eight-modulator  reduced  test  section  configuration. 

duct  and  the  positive  z-direction  is  defined  from  the  left  sidewall  of  the  duct  as 
one  looks  downstream. 


Table  1:  Kulite  (K)  and  microphone  (M)  locations  of  acoustic  measurements. 


Loc. 

Description 

Type 

Coordinate  (m) 

1 

Test  Sect.  Horizontal  Centerline  Upstream 

K 

7.75,  0,  0 

2 

Test  Sect.  Horizontal  Centerline  Downstream 

K 

8.71,  0,  0 

5 

Test  Sect.  Vertical  Centerline  Top 

M 

8,23,  0.3,  0 

15 

Test  Sect.  HorizontaWertical  Centerline 

K 

8,23, 0,  0 

25 

Test  Sect.  Vertical  Centerline  Bottom 

M 

8.23,  -0.3,  0 

28 

2x4  HorizontaWertical  Centerline 

M 

2.19,  0,0 

29 

4x8  Horizontal  Centerline,  %  Downstream 

M 

3.66, 0,  0 

30 

Horn  Tran,  Hor.  Centerline,  %  Downstream 

M 

4.75, 0,  0 

35 

Termination  HorizontaWertical  Centerline 

M 

12.46, 0,  0.17 

The  acoustic  pressure  at  location  1  was  used  as  a  reference  measurement  for 
shaping  the  input  spectrum  and  for  establishing  the  nominal  overall  sound 
pressure  level  for  each  test  condition.  For  each  configuration,  the  input 
spectrum  to  the  air  modulators  was  manually  shaped  through  frequency 
equalization  to  produce  a  nearly  flat  spectrum  at  the  reference  pressure 
transducer.  Data  was  acquired  at  the  noise  floor  level  (flow  noise  only)  and  at 
overall  levels  above  the  noise  floor  in  6  dB  increments  (as  measured  at  the 
reference  location)  up  to  the  maximum  achievable.  Thirty-two  seconds  of 
time  data  were  collected  at  a  sampling  rate  of  4096  samples/s  for  each 
transducer  in  each  test  condition.  Post-processing  of  the  time  data  was 
performed  to  generate  averaged  spectra  and  coherence  functions  with  a  1-Hz 
frequency  resolution. 


924 


RESULTS 

For  each  facility  configuration,  plots  of  the  following  quantities  are  presented: 
normalized  input  spectrum  to  the  air  modulators,  minimum  to  maximum 
sound  pressure  levels  at  the  reference  location,  maximum  sound  pressure 
levels  in  the  test  section,  maximum  sound  pressure  levels  upstream  and 
downstream  of  the  test  section,  and  vertical  and  horizontal  coherence  in  the 
test  section.  The  minimum  levels  in  each  case  correspond  to  the  background 
noise  produced  by  the  airflow  through  the  modulators. 

Normalized  input  voltage  spectra  to  each  modulator  for  each  configuration  are 
shown  in  Figures  10,  15,  20,  25  and  30.  These  spectra  were  generated  to 
achieve  as  flat  an  output  spectrum  as  possible  at  the  reference  location  for  the 
frequency  range  of  interest  (40-200  Hz  for  the  full  section,  40-500  Hz  for  the 
reduced  section).  As  expected,  the  significant  difference  between  the  full  and 
reduced  configurations  is  seen  in  the  high  (>200  Hz)  frequency  content. 

Figure  11  shows  a  background  noise  level  of  126  dB  (the  lowest  of  all 
configurations)  for  the  two-modulator  full  test  section  configuration.  Nearly 
flat  spectra  are  observed  below  210  Hz  for  levels  above  130  dB,  giving  a 
dynamic  range  of  about  32  dB.  The  flat  spectrum  shape  is  a  significant 
improvement  over  the  performance  of  the  old  configuration  as  shown  in  Figure 
2.  Standing  waves  are  evident  at  frequencies  of  210,  340  and  480  Hz.  For  this 
reason,  the  full  section  operation  is  limited  to  less  than  210  Hz  or  to  the  220- 
330  and  370-480  Hz  frequency  bands.  The  effect  of  standing  waves  are 
explored  in  further  depth  in  the  next  section.  The  spectra  in  Figure  12  indicate 
a  nearly  uniform  distribution  in  the  x-direction  throughout  the  test  section.  It  is 
interesting  to  note  that  Figure  13  shows  no  sign  of  standing  waves  upstream  of 
the  test  section,  confirming  that  the  cause  is  associated  with  the  test  section. 
Lastly,  a  near  perfect  coherence  between  upstream  and  downstream,  and  upper 
and  lower  test  section  locations  is  shown  in  Figure  14  for  frequencies  between 
40  and  210  Hz.  Again,  this  is  a  significant  improvement  over  the  performance 
of  the  old  configuration  (Figure  3). 


Figure  10:  Normalized  input  spectrum  Figure  11:  Min  to  max  SPL  at  location 
(2-modulator  full).  1  (2-modulator  full). 


925 


Figure  12:  SPL  in  test  section  at  max  Figure  15:  Normalized  input  spectrum 
level  (2-modulator  full).  (4-modulator  full). 


Figure  13:  SPL  along  length  of  TAFA  Figure  16:  Min  to  max  SPL  at  location 
(2-modulator  full).  1  (4-modulator  full). 


Figure  14:  Test  section  coherence  (2-  Figure  17:  SPL  in  test  section  at  max 
modulator  full).  level  (4-modulator  full). 

The  four-modulator  full  configuration  exhibits  similar  behavior  as  the  two- 
modulator  full  configuration  as  seen  in  Figures  16-19.  The  lowest  level  at 
which  a  uniform  spectrum  is  achieved  is  137  dB,  giving  a  dynamic  range  of 
roughly  30  dB  in  this  configuration.  Lastly,  the  eight-modulator  full 


926 


configuration  results,  shown  in  Figures  21-24,  indicate  a  noise  floor  of  about 
142  dB  and  dynamic  range  of  22  dB. 


Frequency,  Hz  Frequency,  Hz 


Figure  18:  SPL  along  length  of  TAFA  Figure  21:  Min  to  max  SPL  at  location 
(4-modulator  full).  1  (8-modulator  full). 


Frequency,  Hz  Frequency,  Hz 

Figure  19:  Test  section  coherence  (4-  Figure  22:  SPL  in  test  section  at  max 
modulator  full).  level  (8-moduiator  full). 


Frequency,  Hz  Frequency,  Hz 


Figure  20:  Normalized  input  spectrum  Figure  23:  SPL  along  length  of  TAFA 
(8-modulator  full).  (8-modulator  full). 


927 


Frequency,  Hz  Frequency,  Hz 


Figure  24:  Test  section  coherence  (8-  Figure  27:  SPL  in  test  section  at  max 
modulator  full).  level  (2-modulator  reduced). 


Figure  25:  Normalized  input  spectrum  Figure  28:  SPL  along  length  of  TAFA 
(2-modulator  reduced).  (2-modulator  reduced). 


Figure  26:  Min  to  max  SPL  at  location  Figure  29:  Test  section  coherence  (2- 
1  (2-modulator  reduced).  modulator  reduced). 

The  reduced  test  section  configurations  are  used  to  increase  the  frequency 
range  and  maximum  sound  pressure  level  in  the  test  section.  Results  for  the 
two-modulator  reduced  configuration,  shown  in  Figures  26-29,  indicate  a 
nearly  flat  spectrum  between  40  and  480  Hz,  a  noise  floor  of  129  dB  and  a 
dynamic  range  of  about  28  dB.  Coherence  in  the  test  section  is  nearly  unity 


928 


over  this  frequency  range.  This  represents  a  significant  improvement  over  the 
old  facility  configuration.  Results  of  similar  quality  indicate  a  d5mamic  range 
of  roughly  26  and  29  dB  for  the  four-  (Figures  31-34)  and  eight-modulator 
(Figures  36-39)  configurations,  respectively.  Note  that  the  coherence  for  these 
configurations  is  slightly  reduced  at  the  high  frequencies,  but  is  still  very  good 
out  to  480  Hz. 


Figure  30:  Normalized  input  spectrum  Figure  33:  SPL  along  length  of  TAFA 
(4-modulator  reduced).  (4-modulator  reduced). 


160  r 


80 


OASPLs:  134.1,  135.5,  142.1,  147.6, 
153.9,  160.1,  165.9, 167.9 


1.0 

!  1 

1  1 

Loc  1,  Loc  2 
Loc  5,  Loc  25 


400 


100 


200  300 

Frequency,  Hz 


400 


500 


Figure  3 1 :  Min  to  max  SPL  at  location  Figure  34:  Test  section  coherence  (4- 
1  (4-modulator  reduced).  modulator  reduced). 


Figure  32:  SPL  in  test  section  at  max  Figure  35:  Normalized  input  spectrum 
level  (4-modulator  reduced).  (8-modulator  reduced). 


929 


80 


OASPLs;  134.1, 135.7,  141.9, 148.3, 
154.1.  160.0,  165.9. 170.5 


80 


Loc30 

Loci 


100  200  300  400  500  100  200  300  400  500 

Frequency.  Hz  Frequency,  Hz 

Figure  36:  Min  to  max  SPL  at  location  Figure  38:  SPL  along  length  of  TAFA 
i  (8-modulator  reduced).  (8-modulator  reduced). 


Figure  37:  SPL  in  test  section  at  max  Figure  39:  Test  section  coherence  (8- 
level  (8-modulator  reduced).  modulator  reduced). 


Table  2  presents  a  summary  of  the  maximum  average  OASPL  for  each  facility 
configuration.  In  each  case,  the  number  of  active  modulators  were  run  at 
maximum  power  as  an  independent  group  (independently  for  the  single 
modulator  case)  and  the  results  averaged.  For  example,  results  for  one  active 
modulator  were  obtained  by  running  each  modulator  individually  and 
averaging  the  resulting  pressures. 


Table  2:  Summary  of  maximum  average  overall  sound  pressure  levels  (dB). 


Number  of  Active  Modulators 

1 

2 

4 

8 

2-Modulator  Red. 

1 

1  j 

2-Modulator  Full 

i 

4-Modulator  Red. 

159.1 

— 

MSM 

4-Modulator  Full 

155.6 

161.2 

8-Modulator  Red. 

158.4 

— 

mmm 

171.7* 

8-Modulator  Full 

153.0 

158.4 

164.5 

170.0 

^Pressure  scaled  by  ^7?  from  7-modu 

ator  run 

DISCUSSION 


In  this  section,  limiting  behaviors  of  the  full  and  reduced  test  section 
configurations  are  explored  and  the  effect  of  test  section  inserts,  modulator 
coupling  and  wave  tube  performance  are  discussed. 

Limiting  Behaviors 

The  auto-spectra  from  the  full  test  section  configurations  exhibit  sharp 
reductions  in  level  at  approximately  210,  340,  and  480  Hz.  This  behavior 
corresponds  to  measurements  near  nodes  of  vertical  (height)  standing  waves  in 
the  test  section  portion  of  the  wave  tube.  Table  3  summarizes  theoretical, 
resonant  frequencies  and  corresponding  modal  indices  of  the  test  section  duct 
resonances  within  the  excitation  bandwidth.  The  modal  indices  m  and  n 
correspond  to  half  wavelengths  in  the  vertical  and  transverse  (width)  directions 
of  the  cross  section,  respectively.  There  are  several  resonances  that  may  be 
excited  below  500  Hz,  but  only  three  of  these  appear  to  be  significant  at  the 
test  section  transducer  locations  (about  the  horizontal  centerline).  Because  of 
the  presence  of  air  flow  in  the  facility  and  lack  of  measurements  in  the  cross 
section,  it  is  difficult  to  correlate  the  experimental  and  theoretical  modes. 
Measurements  of  the  acoustic  pressure  at  several  locations  in  a  cross-section  of 
the  duct  will  be  necessary  to  fully  characterize  the  resonant  behavior.  It  is 
sufficient  to  say  that  the  usable  frequency  range  in  the  full  test  section 
configurations  is  approximately  40-210  Hz  near  the  horizontal  centerline. 
Acoustic  pressure  auto-spectra  from  the  reduced  test  section  configurations  are 
essentially  flat  to  almost  500  Hz.  This  is  due  to  the  fact  that  only  two 
resonances  are  within  the  excitation  bandwidth  for  this  configuration,  see 
Table  3.  A  sharp  reduction  is  noted  in  the  vicinity  of  480  Hz.  Although  the 
(m=l,  n=0)  resonance  does  not  appear  to  be  significant,  close  inspection  of  the 
data  (not  shown)  indicates  its  presence.  Therefore,  the  usable  frequency  range 
for  the  reduced  test  section  configurations  is  approximately  40-500  Hz. 

Table  3:  Theoretical  resonant  frequencies  of  test  section  duct  modes  in  Hz. 


Performance  of  Test  Section  Configurations 

For  constant  input  acoustic  power,  the  change  from  full  to  reduced  test  section 
configurations  should  theoretically  result  in  a  4,7  dB  increase  in  OASPL. 
However,  Table  2  shows  that  increases  of  only  2.1  (e.g.  164.3-162.2),  0.9,  and 
1.7  dB  were  realized  for  the  two-,  four-  and  eight-modulator  configurations. 
The  system  efficiency  (actual/expected  mean-square  pressure)  of  the  two-, 
four-  and  eight-modulator  reduced  configurations  is  38,  40  and  44  percent, 
respectively,  compared  with  51,  63  and  62  percent  for  the  two-,  four-  and 
eight-modulator  full  configurations.  The  expected  pressure  is  calculated  based 
upon  a  input-scaled  value  of  the  rated  acoustic  power  of  the  WAS  3000 
modulator  assuming  incoherent  sources  (3  dB  per  doubling).  In  general,  the 
full  section  efficiency  is  greater  than  the  corresponding  reduced  section 
efficiency.  While  the  reason  for  this  phenomena  is  not  known,  it  is 
conjectured  that  the  lack  of  expansion  in  the  reduced  configurations  limits  the 
development  of  plane  waves.  Therefore,  phase  and  amplitude  mismatches 
between  acoustic  sources  may  be  accentuated. 

Modulator  Coupling  Performance 

A  simplified  waveguide  analysis  for  coherent,  phase-matched  sources  predicts 
increases  in  OASPL  as  shown  in  Table  4.  Measured  performance  gains  were 
less  than  predicted  because  of  the  assumptions  of  the  waveguide  analysis 
(inactive  source  area  treated  as  hard  wall),  and  possible  reductions  due  to 
phase  differences  between  modulators  and  non-parallel  wave  fironts  at  the  exit 
of  the  modulator  cart,  see  Figures  4-9.  The  latter  effect  is  due  to  different 
angles  of  inclination  of  the  sources  relative  to  the  axis  of  the  wave  tube.  The 
greater  gains  achieved  in  the  full  test  section  configurations  support  the  above 
contention  that  they  are  more  efficient  than  the  reduced  configurations  in 
combining  the  acoustic  sources. 


Table  4:  Change  in  SPL  (dB)  from  1  to  max.  number  of  active  modulators. 


Configuration 

A  SPL  fi:om  1  Active  Mod.  (Meas/Pred) 

2-Modulator  Red.  (2  active  mods.) 

3.9/6.53 

2-Modulator  Full  (2  active  mods.) 

5.5/6.53 

4-Modulator  Red.  (4  active  mods.) 

8.8  / 13.98 

4-Modulator  Full  (4  active  mods.) 

11.4/13.98 

8-Moduiator  Red.  (8  active  mods.) 

13.3/22.10 

8-Moduiator  Full  (8  active  mods.) 

17.0/22.10 

Wave  Tube  Performance 


A  change  in  configuration  from  the  two-  to  the  four-modulator  configurations, 
and  from  the  four-  to  the  eight-modulator  configurations,  will  result  in  an 
incremental  increase  of  3  dB  in  OASPL  if  the  individual  sources  are  phase- 
matched.  This  is  due  to  a  pure  doubling  of  the  power  without  any  change  in 
the  radiation  impedance  of  the  individual  sources.  For  the  reduced 
configurations,  a  3.6  and  3.8  dB  increase  are  observed,  respectively.  A  4.8  and 


932 


3.0  dB  increase  are  observed  for  the  full  configurations,  respectively.  Note 
that  a  greater  than  3  dB  increase  is  possible  when  the  higher  modulator 
configuration  (for  example,  the  four-modulator  reduced  configuration)  is  less 
susceptible  than  the  lower  modulator  configuration  (the  two-modulator 
reduced  configuration)  to  phase  mismatches  between  modulators.  This  seems 
plausible  because  any  such  mismatches  are  averaged  over  a  larger  number  of 
sources. 

SUMMARY 

Modifications  to  the  NASA  Langley  TAFA  facility  resulted  in  significant 
improvements  in  the  quality  and  magnitude  of  the  acoustic  excitation  over  the 
previous  facility.  The  maximum  OASPL  was  increased  by  over  6  dB  (vs  the 
previous  165  dB)  with  a  nearly  flat  spectrum  between  40-210  and  40-480  Hz 
for  the  full  and  reduced  test  section  configurations,  respectively.  In  addition, 
the  coherence  over  the  test  section  was  excellent.  These  improvements, 
however,  did  not  meet  the  objective  for  a  maximum  OASPL  of  178  dB. 

There  are  several  reasons  why  the  maximum  OASPL  did  not  meet  the 
objectives,  including  a  lack  of  expansion  in  the  reduced  configurations  and 
phase  differences  between  modulators.  A  detailed  computational  analysis 
would  be  desirable  to  indicate  the  source  of  the  inefficiencies  and  to  help 
identify  possible  means  of  increasing  the  overall  system  performance. 
ACKNOWLEDGEMENTS 

The  authors  wish  to  thank  Mr.  H.  Stanley  Hogge  and  Mr.  George  A.  Parker  for 
their  support  in  configuring  and  running  the  facility.  We  wish  to  also  thank 
Mr.  James  D.  Johnston,  Jr.  of  NASA  Johnson  Space  Center  for  loan  of  four 
Wyle  air  modulators. 

REFERENCES 

1.  Maestrello,  L.,  Radiation  from  a  Panel  Response  to  a  Supersonic 
Turbulent  Boundary  Layer,  Journal  of  Sound  and  Vibration,  1969, 
10(2),  pp.  261-295. 

2.  Pozefsky,  P.,  Blevins,  R.D.,  and  Langanelli,  A.L.,  Thermal-Vibro- 
Acoustic  Loads  and  Fatigue  of  Hypersonic  Flight  Vehicle  Structure, 
AFWAL-TR-89-3014, 

3.  Williams,  L.J.,  HSCT  Research  Gathers  Speed,  Aerospace  America, 
April  1995,  pp.  32-37. 

4.  Clevenson,  S.A.  and  Daniels,  E.F.,  Capabilities  of  the  Thermal 
Acoustic  Fatigue  Apparatus,  NASA  TM  104106,  February  1992. 


933 


SONIC  FATIGUE  CHARACTERISTICS  OF  HIGH  TEMPERATURE  MATERIALS  AND 
STRUaURES  FOR  HYPERSONIC  FLIGHT  VEHICLE  APPLICATIONS- 


Dr.  I.  Holehouse,  Staff  Specialist, 
Rohr  Inc.,  Chula  Vista,  California 


1.  INTRODUCTION  SUWiARY 

A  combined  analytical  and  experimental  program  was  conducted  to  investigate 
thermal -acoustic  loads,  structural  response,  and  fatigue  characteristics  of 
skin  panels  for  a  generic  hypersonic  flight  vehicle.  Aerothermal  and 
aeroacoustic  loads  were  analytically  quantified  by  extrapolating  existing 
data  to  high  Mach  number  vehicle  ascent  trajectories.  Finite-element 
thermal  and  sonic  fatigue  analyses  were  performed  on  critically  affected 
skin  panels.  High  temperature  random  fatigue  shaker  tests  were  performed 
on  candidate  material  coupons  and  skin-stiffener  joint  subelements  to 
determine  their  random-fatigue  strength  at  high  temperatures.  These  were 
followed  by  high  temperature  sonic  fatigue  tests  of  stiffened-skin  panels 
in  a  progressive  wave  tube.  The  primary  materials  investigated  were 
carbon-carbon  and  silicon-carbide  refractory  composites,  titanium  metal 
matrix  composites  and  advanced  titanium  alloys.  This  paper  reports  on  the 
experimental  work  and  compares  measured  frequencies  and  acoustically 
induced  response  levels  with  analytically  predicted  values. 

The  coupon  shaker  test  data  were  used  to  generate  material  random  fatigue 
"S-N"  curves  at  temperatures  up  to  980°C.  The  joint  subelements  provided 
data  to  determine  the  effects  on  fatigue  life  of  skin-stiffener  joining 
methods.  The  PWT  sonic  fatigue  panel  tests  generated  response  and  fatigue 
life  data  on  representative  built-up  skin  panel  design  configurations  at 
temperatures  up  to  925“C  and  sound  pressure  levels  up  to  165  dB.  These^ 
data  are  used  in  determining  the  response  strains  and  frequencies  of  skin 
panel  designs  when  subjected  to  combined  thermal -acoustic  loading  and  to 
identify  modes  of  failure  and  weaknesses  in  design  details  that  affect _ 
sonic  fatigue  life.  Sonic  fatigue  analyses  of  selected  test  panel  design 
configurations  using  finite-element  techniques  were  also  performed  and 
related  to  the  experimental  results.  Acoustically  induced  random  stresses 
were  analytically  determined  on  a  mode-by-mode  basis  using  finite  element 
generated  mode  shapes  and  an  analytical  procedure  that  extends  Miles 
approach  to  include  multi-modal  effects  and  the  spatial  characteristics  of 
both  the  structural  modes  and  the  impinging  sound  field. 

The  paper  also  describes  the  instrumentation  development  work  performed  in 
order  to  obtain  reliable  strain  measurements  at  temperatures  in  excess  of 
conventional  strain  gauge  capabilities.  This  work  focused  primarily  on  the 
use  of  recently  developed  high  temperature  (350"C  to  1000“C)  strain  gauges, 
laser  Doppler  vibrometers,  high  temperature  capacitance  displacement 
probes,  and  the  determination  of  strain-displacement  relationships  to 
facilitate  the  use  of  double  integrated  accelerometer  data  to  derive  strain 
levels. 


935 


This  work  was  funded  by  the  USAF  Flight  Dynamics  Laboratory  (Kenneth  R. 
Wentz,  Project  Engineer).  The  complete  program  report  is  contained  in 
References  1  and  2. 


2.  HIGH  TEMPERATURE  STRAIN  MEASUREMENTS 

Conventional  adhesively  bonded  strain  gauge  installations  are  temperature 
limited  to  approximately  350°C.  In  order  to  achieve  strain  measurements  at 
higher  temperatures,  up  to  QSO^C,  ceramic  layers  and  coatings  were  used  to 
both  attach  strain  gauges  and  to  thermally  protect  them.  However,  such 
strain  gauge  installations  are  very  sensitive  to  process  parameters  which 
often  need  varying  depending  upon  the  test  specimen  material.  Coated 
carbon-carbon  is  a  particularly  difficult  material  to  adhere  to  due  to  its 
material  characteristics  and  relatively  rough  surface  texture.  Carbon- 
carbon  also  has  a  near  zero  coefficient  of  thermal  expansion  which  presents 
attachment  and  fixturing  problems  in  a  high  temperature  environment. 

When  high  test  temperatures  either  preclude  or  make  problematic  the  use  of 
strain  gauges,  an  alternative  technique  for  obtaining  strain  levels  is  to 
measure  displacements  and  then  determine  strain  levels  using  strain- 
displacement  ratios.  Strain  is  directly  proportional  to  displacement  for  a 
given  deflected  shape,  or  mode  shape,  regardless  of  changes  in  the  elastic 
modulus  of  the  specimen  material  as  it  is  heated.  Consequently,  if  the 
deflected  shape  does  not  change  significantly  with  temperature,  high 
temperature  test  strain  levels  can  be  determined  from  room  temperature 
strain  and  displacement  measurements  in  combination  with  displacement 
measurements  made  at  the  test  temperature. 

This  measurement  technique  facilitates  the  use  of  non- contacting 
transducers  which  can  be  located  away  from  the  heated  area,  such  as 
capacitance  displacement  probes  or  Laser  Doppler  Vibrometers  (LDV).  LDVs 
actually  measure  surface  velocity  but  their  signal  outputs  can  be  readily 
integrated  and  displayed  as  displacement.  Accelerometers  can  also  be  used 
to  measure  displacement  by  double  integrating  their  signal  output. 

However,  since  accelerometers  require  surface  contact  they  have  to  either 
withstand,  or  be  protected  from,  the  thermal  environment.  When  this  is  not 
readily  achievable,  it  is  sometimes  possible  to  install  an  accelerometer  at 
a  location  on  the  test  specimen  or  fixturing  where  the  temperature  is 
within  its  operating  range,  providing  the  displacement  response  at  the 
point  of  measurement  is  fully  coherent  with  the  strain  response  at  the 
required  location. 

The  displacement  range  limitations  of  the  LDV  and  capacitance  probes 
available  to  the  program  resulted  in  having  to  use  double-integrated 
accelerometer  outputs  to  measure  displacements  at  room  temperature  and  at 
the  test  temperature.  Conventional  strain  gauges  were  used  to  measure 
strains  at  room  temperature.  In  order  to  confirm  that  the  strain- 
displacement  ratios  were  unaffected  by  temperature,  limited  high 
temperature  strains  were  measured  at  temperatures  up  to  980* C.  Once  the 
strain-displacement  ratio  for  a  given  specimen  type  was  determined,  air¬ 
cooled  accelerometers  were  used  to  determine  high  temperature  test  strain 
levels.  The  level  of  measurement  accuracy  of  this  technique  was  estimated 
to  be  within  10  percent. 


936 


The  most  successful  strain  measurements  made  at  980“ C  utilized  a  ceramic 
flame  spray  installation  of  an  HFN  type  free  filament  gauge.  This  gauge 
installation  included  the  use  of  silicon-carbide  (SiC)  cement  as  a  base 
coat  for  the  gage,  applied  over  a  1-inch  square  area  of  a  lightly  sanded 
carbon-carbon  surface  substrate.  Lead  wire  attachments  to  the  gauge  were 
made  with  standard  Nichrome  ribbon  wire  anchored  to  the  specimen  with  SiC 
cement.  With  this  gauge  installation,  it  was  possible  to  make  dynamic 
strain  measurements  for  short  periods  of  time  at  980“C. 


3.  RANDOM  FATIGUE  SHAKER  TESTS 

The  instrumented  test  specimens  were  mounted  in  a  duckbill  fixture  and  the 
specimen/fixture  assembly  then  enclosed  in  a  furnace.  An  opening  in  the 
furnace  allows  the  specimen  tip  to  protrude  out  in  order  to  accorrmodate  the 
air-cooled  tip  accelerometer.  Figure  1  shows  strain  gauge  locations  and 
fixturing  for  material  coupon  and  joint  subelement  specimens. 

The  test  procedure  comprised  a  room  temperature  sine-sweep  in  order  to 
identify  the  fundamental  mode  and  its  natural  frequency,  one-third  octave 
random  loading  at  room  temperature  centered  around  the  fundamental  natural 
frequency  and  one-third  octave  random  endurance  testing  at  the  required 
test  temperature  and  load  level. 

Twelve  inhibited  carbon-carbon  material  coupons  generated  usable  S-N  data, 
eleven  at  980“C  and  one  at  650“C.  S-N  data  points  were  also  generated  at 
980°C  for  two  integral  joint  and  two  mechanically  fastened  joint 
subelements.  Fixturing  problems  and  specimen  availability  limited  the 
number  of  S-N  data  points  generated.  Figure  2  shows  the  random  fatigue  S-N 
data  points  with  joint  subelement  data  points  superimposed.  The  random 
fatigue  endurance  level  for  the  material  coupons,  extrapolated  from  10  to 
10®  cycles,  is  approximately  320  microstrain  rms.  The  integral  joint 
subelements  did  not  fail  at  the  strain  gauge  locations;  consequently,  the 
actual  maximum  strain  levels  were  higher  than  those  shown  on  Figure  2. 
Taking  this  into  account,  it  appears  that  the  integral  joints  have  a 
fatigue  endurance  level  of  greater  than  one-half  of  that  for  the  material 
coupons.  The  mechanically  fastened  joint  subelements  exhibited  fatigue 
strength  comparable  to  that  of  the  material  coupons.  These  results 
indicate  that  carbon-carbon  joints  and  attachments  methods  are  not 
critically  limiting  factors  in  the  structural  applications  of  inhibited 
carbon-carbon.  Figure  3  shows  a  representative  example  of  the  strain 
amplitude  and  peak  strain  amplitude  probability  density  functions  at  room 
temperature  for  a  material  coupon  specimen.  The  "peak"  function  can  be 
seen  to  approximate  a  Rayleigh  distribution,  as  it  should  for  a  Gaussian 
random  process. 

Random  fatigue  S-N  data  were  also  generated  for  enhanced  silicon-carbide 
composites  (SiC/SiC)  including  thermally  exposed  specimens  (160  hours  at 
980“ C),  titanium  metal  matrix  composites  (TMC)  utilizing  Ti  15-3  and  Beta 
21S  titanium  matrix  materials,  titanium  aluminide  (super  alpha  two), 
titanium  6-2-4-2,  titanium  6-2-4-2-$i  (including  thermally  exposed 
specimens)  and  Ti-1100.  The  fatigue  endurance  levels  are  shown  in  Table  1. 
Also  shown  in  Table  1  are  S-N  data  points  for  uninhibited  carbon-carbon 
generated  on  a  previous  program  (Reference  3). 


937 


TABLE  1.  SUMMARY  OF  RANDOM  FATIGUE  ENDURANCE  LEVELS. 


material 

TEMPERATURE 

ENDURANCE  LEVEL  CORRESPONDING  TO  10® 
CYCLES:  OVERALL  RMS  STRAIN  (MICROSTRAIN) 

material  COUPONS 

SUBELEMENTS 

INHIBITED  CARBON-CARBON 

1800“F  (980"C) 

320 

integral  JOINTS  >  160 

BOLTED  JOINTS  320 

5  PLY 

1800T  (980"C) 

100 

- 

UNINHlbl 1 tU 
CARBON-CARBON 

1ft  PI  Y 

ISOO'-F  igSOT) 

150 

_ 

*5  PLY 

1000‘'F  I540"C) 

100 

- 

18  PLY 

lOOO'F  (540-C) 

450 

- 

5  PLY 

ROOM  TEMPERATURE 

550 

- 

18  PLY 

ROOM  TEMPERATURE 

450 

- 

ENHANCED 

SiC/SiC 

NON-EXPOSED 

1800"F  r980“C) 

450 

- 

THERMALLY  EXPOSED 

iaOO“F  f980‘'C) 

300 

- 

1000“F  (540*0 

520 

DIFFUSION-BONDED  HAT- 
STIFFENED  =  520 

n  IML 

ROOM  TEMPERATURE 

2250 

- 

beta  21S  TMC 

200 

510 

- 

TITANIUM  ALUMINIDE 

/ciinCD  ftl  DUIA  TUn^ 

ROOM  TEMPERATURE 

410 

{  rvui  1  iri  •  * 

Ti-6242-Si 

735 

Hilllll 

LIO  BONDED  HONEYCOMB 
BEAM  =  388 

■■ 

WELDED  JOINT  =  400 

T-;  nnn 

- 

Ti  6-2-4-2 

ROOM  TEMPERATURE 

675 

- 

866MISC/039-T1.IH 

12-02-96 


938 


Figure  4  shows  random  fatigue  S-N  curves  for  the  materials  tested 
superimposed  on  one  graph  for  comparison  purposes.  The  Ti  15-3  TMC  data 
are  not  shown  since  this  was  a  concept  demonstrator  material  utilizing  a 
Ti  15-3  matrix  material  for  producibility  reasons.  Ti  15-3  does  not  have 
the  temperature  capability  for  hypersonic  vehicle  applications.  Titanium 
aluminide  data  are  not  shown  due  to  its  brittle  material  characteristics 
making  it  unsuitable  for  sonic  fatigue  design  critical  structures.  Ti-1100 
S-N  data  were  very  similar  to  the  non-exposed  Ti  6-2-4-2-Si  and  are  not 
shown.  Ti  6-2-4-2  coupons  were  only  tested  at  room  temperature  before 
beingVeplaced  by  Ti  6-2-4-2-Si,  which  has  higher  structural  temperature 
capabilities. 

The  fatigue  curves  in  Figure  4  show  inhibited  carbon-carbon  to  have  higher 
fatigue  strength  at  980°C  than  does  its  uninhibited  counterpart.  Inhibited 
carbon-carbon  also  has  greater  resistance  to  oxidation  at  high 
temperatures. 

Although  unexposed  enhanced  SiC/SiC  had  greater  random  fatigue  strength  at 
980“C  than  did  inhibited  carbon-carbon,  the  two  materials  exhibited  similar 
strength  at  temperature  after  allowing  for  thermal  exposure.  However, 
SiC/SiC  has  a  maximum  temperature  capability  of  1100  to  1200°C  compared  to 
1700  to  1900“C  for  carbon-carbon. 

The  Beta  21S  TMC  material  demonstrated  resonable  fatigue  strength  at  815°C 
and  the  Ti  6-2-4-2-Si  specimens  exhibited  high  fatigue  strength  at  620°C  to 
650“C. 


4.  SONIC  FATIGUE  PANEL  TESTS 

These  tests  were  performed  in  Rohr's  high  temperature  progressive-wave  tube 
(PWT)  test  facility.  The  facility  is  capable  of  generating  overal 1^ sound 
pressure  levels  of  165  to  168  dB  at  temperatures  up  to  925 “C  to  980“ C, 
depending  upon  the  test  panel  configuration  and  material. 

Three  rib-stiffened  carbon-carbon  panels  and  a  monolithic  hat-stiffened 
Beta  21S  TMC  panel  were  subjected  to  sonic  fatigue  testing.  Response 
strains  were  measured  on  the  four  panels  over  a  range  of  incrementally 
increasing  sound  pressure  levels  (140  to  165  dB)  at  room  temperature.  One 
carbon-carbon  panel  was  subjected  to  sonic  fatigue  testing  at  room 
temperature  and  the  other  two  tested  at  925“ C.  The  TMC  panel  was  endurance 
tested  at  815“C.  Figures  5  and  6  show  a  carbon-carbon  panel  and  its 
fixturing  installed  in  the  PWT.  The  panels  were  attached  to  the  fixture 
via  flexures  in  order  to  allow  for  differences  in  the  thermal  expansion  of 
the  panel  and  fixture  materials.  Structural  details  of  the  panels  and 
instrumentation  locations  are  given  in  References  1  and  2. 

The  three  carbon-carbon  panel  configurations  encompassed  two  skin 
thicknesses  and  two  stiffener  spacings  as  follows: 

Panel  1:  3  skin  bays,  6  in.  by  20  in.  by  0.11  in.  thick 

Panel  2:  2  skin  bays,  9  in.  by  20  in.  by  0.11  in.  thick 

Panel  3:  3  skin  bays,  6  in.  by  20  in.  by  0.17  in.  thick 


939 


Table  2  summarizes  the  measured  room  temperature  frequencies  and  strain 
response  levels: 


TABLE  2.  ROOM  TEMPERATURE  RESPONSE  OF  TEST  PANELS- 


TEST 

PANEL 

FREQUENCY  OF 
IN-PHASE 
MODE  (Hz) 

OASPL 

(dB) 

OVERALL  RMS  STRAINS 
rMICROSTRAIN) 

EDGE  OF 
SKIN  BAY 

CENTER  OF 
SKIN  BAY 

CARBON-CARBON  NO.  1 

267 

165 

305 

149 

155  &  171 

145 

126 

39 

191 

59 

165 

558* 

173* 

CARBON-CARBON  NO.  3 

423 

165 

69 

127 

BETA  21$  TMC 

241 

165 

HIGHEST  STRAIN  =  287 

AT  PANEL  CENTER  ON 

STIFFENER  CAP 

*  EXTRAPOLATED  ON  THE  BASIS  OF  TUE  STRAIN  RESPONSE  WITH  SPL  FOR 
PANELS  1  AND  3. 


Panel  1  was  subjected  to  165  dB  at  room  temperature  for  10  hours  at  which 
point  cracks  developed  at  the  ends  of  the  stiffeners.  The  frequency 
dropped  slightly  during  the  ten  hour  test  resulting  in  the  number  of  cycles 
to  failure  being  approximately  9  million. 

Panel  2  was  endurance  tested  at  925“C  at  150,  155  and  160  dB  for  3-1/2 
hours  at  each  level,  followed  by  one  hour  at  165  dB.  At  this  point,  cracks 
were  observed  at  the  ends  of  the  stiffeners,  similar  to  the  cracks  in 
Panel  1. 

Panel  3  was  endurance  tested  at  925“C  and  165  dB  for  10  hours  without  any 
damage  to  the  panel. 

The  TMC  panel  was  endurance  tested  at  815° C  and  165  dB  for  3-1/2  hours  at 
which  time  cracks  were  observed  in  two  stiffener  caps  at  the  panel  center. 

The  high  test  temperatures  for  Panels  2  and  3  and  the  TMC  panel  precluded 
attaching  an  accelerometer  directly  to  the  panel  surface,  even  with  air 
cooling.  This  prevented  the  direct  measurement  of  panel  displacements  at 
925° C.  In  order  to  attempt  to  estimate  the  high  temperature  endurance  test 
strain  levels,  a  temperature  survey  was  performed  on  the  panel  fixturing 
with  Panel  3  installed  in  order  to  determine  an  acceptable  location  for  an 
accelerometer.  An  accelerometer  at  the  selected  fixture  location  tracked 
linearly  with  the  highest  reading  strain  gauges  during  a  room  temperature 
response  survey.  The  coherence  between  the  fixture  accelerometer  and  the 
panel  strain  gauges  was  0.9  in  the  frequency  range  of  panel  response. 


940 


Having  established  a  coherent  strain  displacement  relationship  at  room 
temperature,  the  temperature  was  increased  progressively  with  increasing 
acoustic  loading,  generating  accelerometer  and  microphone  data  at  480  C  and 
140  dB,  650°C  and  155  dB,  860°C  and  155  dB,  and  925“C  at  165  dB.  It  was 
clear  from  the  data  at  the  higher  temperatures  and  load  levels  that  the 
full  spectrum  overall  rms  displacement  levels  obtained  by  double  integrat¬ 
ing  the  accelerometer  output  signals  could  not  be  used  to  determine  high 
temperature  strain  levels  due  to  high  amplitude,  low  frequency  displace¬ 
ments  (displacement  being  inversely  proportional  to  frequency  squared  for  a 
given  "g"  level)  that  were  well  below  the  panel  response  frequency  range 
and  therefore  would  not  be  proportional  to  panel  strain  levels.  It  is 
important  to  remember  here  that  since  the  accelerometer  is  mounted  on  the 
panel  fixture,  it  is  measuring  fixture  response,  some  of  which  is  not 
related  to  panel  response. 


After  reviewing  the  various  frequency  spectra,  it  was  decided  to  re-analyze 
the  data  to  generate  overall  rms  levels  over  selected  frequency  bandwidths 
that  would  encompass  a  high  percentage  of  the  full -spectrum  overall  rms 
strains  and  eliminate  the  low  frequency  displacements.  If  a  consistent 
strain-displacement  relationship  could  be  established  at  room  temperature 
within  a  frequency  bandwidth  such  that  the  strains  could  be  related  to  the 
full-spectrum  overall  rms  strains,  and  if  the  same  bandwidth  could  be  used  to 
generate  displacements  at  temperatures  that  were  sufficiently  consistent  to 
relate  to  strain  response,  then  it  would  be  possible  to  at  least  make  a 
reasonable  estimate  of  the  test  temperature  strain  level.  It  was  determined 
that  band-passed  response  data  in  the  300  to  600  Hz  frequency  range  gave 
consistent  strain-displacement  ratios  at  room  temperature.  Double-integrated 
band-passed  accelerometer  outputs  (displacements)  were  consistent  with 
increasing  sound  pressure  levels  at  incrementally  increasing  test  temperatures 
up  to  the  925°C/165  dB  endurance  test  conditions.  Table  3  summarizes  the  high 
temperature  test  panel  results. 


TABLE  3.  HIGH  TEMPERATURE  TEST  PANEL  RESULTS. 


TEST  PANEL 

TEST 

TEMPERATURE 

OVERALL 
SOUND 
PRESSURE 
LEVEL  (dB) 

HIGHEST 
ESTIMATED 
OVERALL  RMS 
STRAIN 

(MICROSTRAIN) 

EXPOSURE  TIME, 
ESTIMATED  FATIGUE 

CYCLES  AND  COMMENTS 

(°F) 

rc) 

BETA  21S  TMC 
PANEL 

1500 

815 

165 

NOT  ESTIMATED 

3  1/2  HRS,  3x10“  CYCLES, 
STIFFENERS  CRACKED  AT 
MID-SPAN 

CARBON-CARBON 
PANEL  NO.  2 

150 

155 

TTTORSTTTMo^TYOlsr 

NO  FAILURE 

155 

219  ^ 

3  1/2  HRS,  2.3x10°  CYCLtS, 
NO  FAILURE 

160 

316 

3  1/2  HRS.  2.3x10“  CYCLES, 
NO  FAILURE 

165 

453 

1  HR,  6.4xl0‘>  cycles! 

CRACKS  AT  STIFFENER  ENDS 

925 

165 

103 

10  HRS,  1.7x10'  CYCLES, 

NO  FAILURE 

941 


It  should  be  noted  that  carbon-carbon  panels  1  and  2  exhibited  cracks  at 
the  stiffener  ends,  whereas  the  maximum  measured  strains  were  at  the  edges 
of  skin  bays.  Consequently,  the  actual  strain  levels  at  the  crack 
locations  were  either  higher  than  the  measured  levels  or  there  were 
significant  stress  concentrations  at  the  stiffener  terminations. 


5.  COMPARISON  OF  ANALYTICAL  AND  TEST  RESULTS  FOR  CARBON-CARBON  PANELS 

MSC  NASTRAN  was  used  to  perform  finite  element  analyses  on  the  three 
carbon-carbon  panels  that  were  subjected  to  the  sonic  fatigue  testing 
described  in  Section  4.  The  oxidation  resistant  coating  was  modeled  as  a 
non-structural  mass,  which  is  compatible  with  the  panel  test  results. ^ 
Natural  frequencies,  mode  shapes  and  acoustically  induced  random  strain 
levels  were  analytically  determined  for  room-temperature  conditions  and 
compared  to  the  room-temperature  panel  test  results. 

Acoustically  induced  random  stresses  were  analytically  determined  on  a 
mode-by-mode  basis  using  the  finite  element  generated  mode  shapes  and  a 
Rohr  computer  code  based  on  an  analytical  procedure  presented  in 
Reference  4.  This  procedure  extends  Miles*  approach  (Reference  5)  to 
include  multi-modal  effects  and  the  spatial  characteristics  of  both  the 
structural  modes  and  the  impinging  sound  field. 

Table  4  shows  the  calculated  and  measured  frequencies,  overall  rms  strain 
levels  and  the  strain  spectrum  levels  for  the  in-phase  stiffener  bending 
mode  for  the  carbon-carbon  panels  at  room  temperature. 


TABLE  4.  CALCULATED  AND  MEASURED  RESPONSE  FREQUENCIES  AND  STRAIN 
LEVELS  FOR  CARBON-CARBON  PANELS  AT  ROOM  TEMPERATURE. 


STRAIN  LEVELS  AT  EDGE  OF  SKIN  BAY 

NATUF 
FREQUE 
OF  IN-F 
MODE 
(Hzl 

\fKL 

:ncy 

>HASE 

OVERALL  RMS  STRAIN 
(MICROSTRAIN) 

STRAIN 

SPECTRUM  LEVEL 
IN-PHASE  MODE 
(MICROSTRAIN/Hz) 

FE  ANALYSIS 

MEASURED 

FE  ANALYSIS 

MEASURED 

FE  ANALYSIS 

MEASURED 

PANEL  1 
a65  dBl 

305 

267 

510 

305 

84 

60 

PANEL  2 
(145  dBl 

190 

155  & 
171 

133 

126 

40 

41 

PANEL  3 
(165  dB) 

460 

423 

77 

69 

16 

16 

942 


The  above  results  show  good  agreement  between  the  finite  element  generated 
values  and  those  measured.  The  level  of  agreement  is  particularly  good  for 
the  strain  spectrum  levels,  which  are  typically  more  difficult  to 
accurately  predict.  Figure  7  shows  the  finite-element  frequency  solution 
for  Panel  3.  The  in-phase  mode  shape  can  be  seen  to  have  an  overall  modal 
characteristic  due  to  the  relatively  low  bending  stiffness  of  the 
stiffeners  for  the  skin  thickness  used.  Figure  8  shows  the  measured  and 
finite-element  generated  strain  frequency  spectra  for  Panel  3. 

Details  of  the  finite-element  analyses  and  models  are  contained  in 
References  1  and  2. 


6.  CONCLUSIONS  AND  RECOMMENDATIONS 

1.  The  high  temperature  testing  techniques  and  strain  measuring 
procedures  successfully  generated  usable  random  fatigue  S-N 
curves  and  panel  response  data.  The  use  of  strain-displacement 
ratios  were  shown  to  be  an  effective  alternative  to  high 
temperature  strain  gauge  measurements. 

2.  In  general,  the  materials  and  structural  concepts  tested 
demonstrated  their  suitability  for  hypersonic  flight  vehicle  skin 
panel  applications.  The  major  exception  was  Titanium-Aluminide 
Super  Alpha  Two  which  was  determined  to  be  too  brittle. 

3.  Inhibited  carbon-carbon  exhibited  significantly  higher  random 
fatigue  strength  at  980°C  than  did  the  uninhibited  carbon-carbon 
—  two  to  three  times  the  random  fatigue  endurance  strain  level. 

4.  Thermally  exposed  enhanced  SiC/SiC  had  comparable  fatigue 
strength  to  that  of  inhibited  carbon-carbon  at  980 °C. 

5.  The  TMC  specimens  usefully  demonstrated  the  fatigue  strength  of 
the  TMC  concept  and  the  need  to  develop  the  concept  to 
incorporate  higher  temperature  capability  titanium  matrix 
materials. 

6.  Titanium  6-2-4-2-Si  exhibited  high  fatigue  strength  in  the  590°C 
to  650 “C  temperature  range  and  also  demonstrated  the  need  for  TMC 
materials  to  utilize  higher  temperature  matrix  materials  in  order 
to  be  cost  effective  against  the  newer  titanium  alloys. 

7.  The  level  of  agreement  between  the  finite  element  analysis 
results  for  the  carbon-carbon  panels  and  the  progressive-wave 
tube  test  data  demonstrated  the  effectiveness  of  the  analytical 
procedure  used.  The  analysis  of  structures  utilizing  materials 
such  as  carbon-carbon  clearly  presents  no  special  difficulties 
providing  the  material  properties  can  be  well  defined. 


943 


8  It  is  recommended  that  further  tests  be  conducted  similar  to 
those  performed  in  this  program  but  with  greater  emphasis  on 
testing  panels  having  dimensional  variations  in  order  to  develop 
design  criteria  and  life  prediction  techniques.  Such  testing 
should  be  performed  on  those  structural  materials  and  design 
concepts  that  emerge  as  the  major  candidates  for  flight  vehicle 
applications  as  materials  development  and  manufacturing 
techniques  progress. 

REFERENCES 


1  R  D.  Blevins  and  I.  Holehouse,  "Thermo-Vibro  Acoustic  Loads  and 
rkigue  of  Hypersonic  Flight  Vehicle  Structure,"  Rohr,  Inc. 
Engineering  Report  RHR  96-008,  February  1996. 

2.  United  States  Air  Force  Systems  Command,  Flight  Dynamics 
Laboratory  Final  Technical  Report,  Contract  No.  F33615-87-C-33^^/, 
to  be  published. 

3.  R.  D.  Blevins,  "Fatigue  Testing  of  Carbon-Carbon  Acoustic  Shaker 
Table  Test  Coupons,"  Rohr,  Inc.  Engineering  Report  RHR  91-087, 
September  1991. 

4.  R.  D.  Blevins,  "An  Approximate  Method  for  Sonic  Fatigue  Analysis 
of  Plates  and  Shells,"  Journal  of  Sound  and  Vibration,  Vol.  129, 
51-71,  1989. 

5.  J.  W.  Miles,  "On  Structural  Fatigue  Under  Random  Loading," 

Journal  of  Aeronautical  Sciences,  Vol.  21,  November  1954. 


944 


A.  Test  Configuration  for 
Material  Coupons 


Fixture 


Specimen 
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B.  Test  Configuration  for 
Carbon-Carbon  Integral 
Stiffener  Specimens 


Accelerometer 


t 


C.  Test  Configuration  For 
■  Carbon-Carbon  Mechanically 
Fastened  Stiffener  and  All 
Titanium  Diffusion  Bonded 
Joint  Specimens 


FIGURE  1  Typical  Strain  Gauge  Locations  and  Test  Configurations 
for  Material  Coupon  and  Joint  Subelement  Shaker  Test 
Specimens 


945 


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FIGURE?  Finite  Element  Solution  for  In-Phase  Overall  Mode  — 
Carbon-Carbon  Panel,  Concept  3 


950 


STRRIN  [Ins  xlB^G/Inl 


952 


SCALING  CONCEPTS  IN  RANDOM  ACOUSTIC  FATIGUE 

BY 

Marty  Ferman*  and  Howard  Wolfe** 

ABSTRACT 

Concepts  are  given  for  scaling  acoustic  fatigue  predictions  for  application  to 
extreme  environmental  levels  based  on  testing  “  scaled”  structures  at 
existing,  lesser  environmental  levels.  This  approach  is  based  on  scaling  a  test 
structure  to  fit  within  the  capabilities  of  an  existing  test  facility  to  attain 
fatigue  results,  and  then  using  analytical  extrapolation  methods  for  predicting 
the  full  scale  case  to  achieve  accurate  design  results.  The  basic  idea  is  to 
utilize  an  existing  acoustic  fatigue  facility  to  test  a  structure  which  has  been 
designed  (scaled)  to  fatigue  within  that  facility’s  limits,  employing  the 
appropriate  structural  properties  ( such  as  thinning  the  skins,  etc.).  Then,  the 
fatigue  life  of  the  actual  structure  is  determined  by  analytically  scaling  the  test 
results  to  apply  to  the  full  scale  case  (  thicker)  at  higher  noise  levels  for 
example.  Examples  are  given  to  illustrate  the  approach  with  limits  suggested, 
and  with  the  recognition  that  more  work  is  needed  to  broaden  the  idea. 

BACKGROUND 

While  it  is  important  to  continually  expand  the  capability  of  acoustic  test 
facilities  ,  it  is  perhaps  equally  important  to  be  able  to  work  with  existing 
facilities  at  any  time.  That  is,  facility  expansions,  enhancements  ,  and 
modernization’s  should  always  be  sought  from  time  to  time,  so  long  as 
practical  and  affordable  from  cost  effective  considerations.  Limits  should  be 
pushed  to  accommodate  larger  sizes  of  test  specimens  with  higher  noise  levels 
with  wider  ranges  of  frequencies,  with  wider  ranges  of  temperatures,  and  with 
better  capabilities  for  applying  pressures  along  with  any  one  of  several  types 
of  preloads.  These  are  costly  considerations  and  require  considerable  time  to 
accomplish.  Facility  rental  can  be  used  in  some  cases  to  bolster  one’s  testing 
facilities,  however  if  the  application  suggests  a  situation  beyond  any  available 
facility  for  the  required  design  proof,  then  an  alternate  is  needed.  Thus  the 
scaling  concept  suggested  here  is  a  viable  and  useable  possibility. 

The  Author’s  basis  for  the  approach  stems  from  their  extensive,  collective, 
experience  in  Structural  Dynamics,  especially  work  in  Acoustic  Fatigue, 
Fluid-Structure  Interaction,  Buffet ,  and  Aeroelasticity/Flutter ,  and 
particularly  from  experience  with  flutter  model  testing,  in  which  it  is  quite 
common  to  ratio  test  results  from  a  model  size  to  full  scale  for  valid 


Assoc.  Prof,  Aerospace  and  Mech.  Engr.  Dept,  Parks  College,  St.  Louis 
Univ.,  Cahokia,  IL,  62206,  USA 

**  Aerospace  Engineer,  Wright  Laboratory,  Wright  Patterson  AFB,  OH, 
45433,  USA 


953 


predictions  .  Flutter  is  a  well  recognized  area  where  model  data  is  commonly 
used  in  nondimensional  form  to  establish  design  margins  of  safety,  as  typified 
in  Ref  (1-2).  Flutter  can  be  nondimensionalized  quite  broadly  as  pointed  out 
in  many  works,  and  is  clearly  done  for  a  wide  range  of  general  cases  using  the 
“so-called”  Simplified  Flutter  Concept,  Ref  (3) .  The  degree  of  the  use  of  the 
flutter  model  scaling  rules  varies  considerably  today,  because  some  people 
are  testing  as  much  or  more  than  ever,  while  others  are  testing  less  and  relying 
more  heavily  on  advanced  theories  such  as  Computational  Fluid  Dynamics, 
CFD.  However,  the  basic  ideas  in  flutter  model  scaling  are  still  POWERFUL! 
In  fact,  this  concept  has  fueled  the  Author’s  desires  to  develop  the 
“acoustical  scaling”  used  in  the  approach  presented.  Moreover,  when  starting 
to  write  this  paper,  the  Authors  realized  that  this  type  of  scaling  is  also 
common  to  many  related  areas  of  structural  dynamics,  and  thus  chose  to 
include  some  examples  of  those  areas  to  emphasize  the  main  point  here! 

For  example,  experience  in  fluid-structure  interaction  and  fatigue  of  fuel  tank 
skins,  a  related  work  area,  serves  as  another  example  of  scaling  structures  to 
demonstrate  accurate  predictions  with  widely  varied  environmental  levels, 
and  a  multitude  of  configurations.  Scaling  and  nondimensional  results  were 
used  extensively  in  Ref  (4  -13),  and  are  cited  here  because  of  the  immense 
data  base  accumulated.  The  work  at  that  time  did  not  necessarily  define 
scaling  as  used  here ,  but  hindsight  now  suggests  that  there  is  a  clear  relation. 

It  is  becoming  well  recognized  that  Buffet  is  easily  scaled  ,  and  many 
engineers  and  investigators  are  now  employing  scaling  of  pressures  from 
model  to  full  size  applications,  and  are  also  using  scaled  model  response  to 
predict  full  scale  cases  .  Some  of  the  earliest  and  some  of  the  more  modern 
results  clearly  show  this  aspect.  For  example.  Ref  (14  -16)  are  typical,  quite 
convincing,  and  pace  setting  regarding  scaled  data.  Buffet  models  which  are 
much  more  frail  that  the  full-scale  cases  are  used  to  develop  data  for  full  scale 
applications,  and  besides  giving  full  sized  results,  provide  a  guide  to  safe 
flight  testing  as  has  been  done  more  extensively  with  flutter  testing. 

Obviously,  acoustical  response  and  fatigue  phenomenon  are  also 
nondimensionalizable  and  scaleable.  Ref  (17-18),  for  example.  This  point  is 
being  taken  further  here;  that  is  ,  scaling  will  be  used  to  take  better  advantage 
of  limited  facility  testing  capability  to  predict  more  severe  situations,  as  is 
used  in  the  case  of  flutter  model  testing  where  a  larger  specimen  is  predicted 
from  tests  of  a  smaller  structure  using  similarity  rules.  Here  in  the  acoustic 
application,  a  thinner  ,  or  otherwise  more  responsive  specimen,  is  tested  and 
then  analytical  means  are  used  to  make  the  prediction  for  the  nominal  case. 

APPROACH 

The  method  is  shown  here  is  basically  an  extension  of  the  flutter  model 
scaling  idea,  as  applied  to  acoustical  fatigue  testing  with  a  particular  emphasis 
on  random  applications.  The  technique  will  also  work  for  sine  type  testing  in 


954 


acoustical  fatigue,  and  perhaps  it  will  be  even  more  accurate  there,  but  most  of 
today’s  applications  are  with  random  testing,  notably  in  the  aircraft  field. 
Thus  it  is  in  this  area  where  the  method  should  find  more  application  .  The 
Authors  have  a  combined  professional  work  experience  of  some  70+  years 
and  thus  have  tried  to  focus  this  extensive  background  on  an  area  where  gains 
can  be  made  to  help  reduce  some  costs  while  making  successful  designs,  by 
using  lesser  testing  capability  than  might  be  more  ideally  used.  It  is  believed 
that  the  best  testing  for  random  acoustic  fatigue,  is  of  course,  with  (a)  the 
most  highly  representative  structure,  and  as  large  a  piece  as  can  be  tested, 
both  practically  and  economically,  (b)  the  most  representative  environmental 
levels  in  both  spectrum  shape  and  frequency  content,  (c)  test  times  to 
represent  true  or  scaled  time,  as  commonly  accepted,  (d)  temperatures  should 
be  applied  both  statically  and  dynamically,  and  finally  (e)  preloading  from 
pressures,  vibration,  and  from  boundary  loading  of  adjacent  structure. 
Frequently,  testing  is  done  to  accomplish  some  goal  using  a  portion  of  these 
factors,  and  the  remainder  is  estimated  .  Thus  the  Authors  believed  that  there 
is  a  high  potential  to  extend  the  flutter  model  approach  to  acoustical 
applications. 

Recall  that  in  the  flutter  model  approach  ,  the  full  scale  flutter  speed  is 
predicted  by  the  rule 


((Vf)a)p-  [  ((Vi.)m)e/  ((  Vf)m)c]  X  [  ((Vf)a)c]  (^) 

where  Vp  is  flutter  speed,  the  subscripts  M  and  A  refer  to  model  and  aircraft 
respectively,  the  subscript  C  refers  to  calculated,  and  the  subscript  P  refers  to 
predicted.  Thus  the  equation  suggests  that  the  full  scale  predicted  flutter  speed 
is  obtained  by  taking  the  ratio  of  experimental  to  calculated  flutter  speed  for 
the  model  and  then  multiplying  by  a  calculated  speed  for  the  airplane.  These 
flutter  model  scaling  ideas  are  covered  in  any  number  of  References,  i.e.  Ref 
(lo),  for  example. 

The  same  concept  can  be  utilized  in  acoustic  fatigue,  i.e.  the  strain  at  fatigue 
failure  relation,  (8,N)  can  be  scaled  from  model  structure  tested  at  one  level 
and  then  adjusted  for  structural  sizing  and  environmental  levels.  This  relation 
can  be  addressed  as  done  for  the  flutter  case: 

((e,N)a)p=[((  s,N)m)e  /  ((e,N)m)c]  X  [((e.N)a)c]  (2) 

where  s  is  strain,  and  N  is  the  number  of  cycles  at  failure,  where  as  above  in 
Eq  (1) ,  the  subscripts  M  and  A  refer  respectively  to  Model  and  Full  Scale  for 
parallelism,  while  the  subscripts  E,  C,  and  P  have  the  same  connotation  again, 
namely,  experimental,  calculated,  and  predicted.  Thus  the  full  scale  case  is 


955 


predicted  from  a  subscale  case  by  using  the  ratio  of  experimental  to 
theoretical  model  results  as  adjusted  by  a  full  scale  calculation.  Flutter  model 
scaling  depends  upon  matching  several  nondimensional  parameters  to  allow 
the  scaling  steps  to  be  valid.  While  these  same  parameters  are,  of  course,  not 
necessarily  valid  for  the  acoustic  relationships,  other  parameters  unique  to 
this  acoustical  application  must  be  considered,  and  will  be  discussed. 
Accurate  predictions  for  the  method  relies  on  extensive  experience  with  the 
topic  of  Acoustic  Fatigue  in  general,  because  concern  is  usually  directed 
towards  the  thinner  structure  such  as;  panels,  panels  and  stiffeners,  and  panels 
and  frames,  bays  (a  group  of  panels),  or  other  sub-structure  supporting  the 
panels.  These  structures  are  difficult  to  predict  and  are  quite  sensitive  to  edge 
conditions,  fastening  methods,  damping,  combination  of  static  and  dynamic 
loading,  and  temperature  effects.  Panel  response  prediction  is  difficult ,  and 
the  fatigue  properties  of  the  basic  material  in  the  presence  of  these  complex 
loadings  is  difficult.  However,  the  experienced  Acoustic  Fatigue  Engineer  is 
aware  of  the  limits,  and  nonnally  accounts  for  these  concerns.  Thus  the 
method  here  will  show  that  these  same  concerns  can  be  accounted  for  with  the 
scaling  approach  through  careful  considerations. 

The  Authors  believe  that  the  method  is  best  explained  by  reviewing  the 
standard  approach  to  acoustic  fatigue,  especially  when  facility  limits  are  of 
major  concern.  Fig  (1)  was  prepared  to  illustrate  these  points  of  that 
approach.  Here  it  is  seen  that  key  panels  for  detail  design  are  selected  from  a 
configuration  where  the  combination  of  the  largest,  thinnest,  and  most 
severely  loaded  panels  at  the  worst  temperature  extremes  and  exposure  times 
are  considered.  These  can  be  selected  by  many  means  ranging  from  empirical 
methods,  computational  means,  and  the  various  Government  guides.  Ref  (17- 
18),  for  example.  Then  detailed  vibration  studies  are  run  using  Finite 
elements  ,  Rayleigh  methods.  Finite  Difference  methods,  etc.  to  determine  the 
modal  frequencies  and  shapes,  and  frequently  linearity  is  assessed.  Then 
acoustical  strain  response  of  the  structure  is  determined  for  sine,  narrowband, 
and  broadband  random  input  to  assess  fatigue  life  based  on  environmental 
exposure  times  in  an  aircraft  lifetime  of  usage.  These  theoretical  studies  are 
then  followed  by  tests  of  the  worst  cases,  where  vibration  tests  are  conducted 
to  verify  modal  frequencies,  shapes,  and  damping,  and  linearity  is  checked 
again  for  the  principal  modes.  This  is  followed  by  acoustical  strain  response 
tests  where  the  strain  growth  versus  noise  levels  is  checked,  again  employing 
sine,  narrowband  and  broadband  random  excitation.  Note  the  figure  suggests 
that  data  from  the  vibration  tests  are  fed  back  to  the  theoretical  arena  where 
measured  data  are  used  to  update  studies  and  to  correlate  with  predictions, 
especially  the  effect  of  damping  on  response  and  fatigue,  and  of  course,  the 
representation  of  nonlinearity.  Also,  the  measured  strain  response  is  again 
used  to  update  fatigue  predictions.  These  updates  to  theory  are  made  before 
the  fatigue  tests  are  run  to  insure  that  nothing  is  missed.  However,  in  this 


956 


case,  the  required  sound  presssure  level  SPL  in  (dB)  is  assumed  to  exceed  the 
test  chamber’s  capability.  Thus  ,  as  shown  in  the  sketch  in  Fig  (2)  the  key 
strain  response  curve,  s  vs  dB,  is  extrapolated  to  the  required  dB  level.  This 
data  is  merged  with  the  strain-to-failure  curve  at  the  right  to  establish  the  cycle 
count,  N,  giving  the  (s,  N)  point  for  this  case.  The  extrapolated  data  provides 
some  measure  of  the  estimated  life,  but  again  is  heavily  dependent  upon  the 
accuracy  of  the  basic  strain  response  curve,  and  is  especially  dependent  on 
whether  high  confidence  exists  at  the  higher  strains.  Linear  theory  is  also 
shown  in  this  case,  indicating  it  overpredicts  the  test  strain  response  and  hence 
shows  a  shortened  fatigue  life  compared  to  test  data,  as  is  generally  the  case 
in  today’s  extreme  noise  levels.  This  illustration  is  highly  simplified, 
because  experienced  designers  readily  know  that  it  is  difficult  to  predict  even 
simple  panels  accurately  at  all  times,  let  alone  complex  and  built-up  structure 
consisting  of  bays  (multi-panel);  this  will  addressed  again  later  in  the  paper. 

The  new  concept  of  scaled  acoustic  fatigue  structures  is  shown  on  the  sketch 
of  Fig  (3)  where  the  standard  method  is  again  shown,  but  smaller  in  size,  to 
refocus  the  thrust  of  the  new  idea.  The  scaling  process  parallels  the  standard 
approach,  and  actually  complements  that  approach,  so  that  the  two  can  be  run 
simultaneously  to  save  time,  costs,  and  manpower.  Here  the  panel  selection 
process  recognizes  that  the  design  application  requires  environments  far  in 
excess  of  available  facility  capability.  Thus  the  scaling  is  invoked  in  the 
beginning  of  the  design  cycle.  As  the  nominal  panels  (bays)  are  selected  and 
analyzed  for  vibration,  response  and  fatigue,  scaled  structures  are  defined  to 
provide  better  response  within  the  existing  chamber  ranges  so  that  they  can  be 
fatigued  and  then  the  results  can  be  rescaled  to  the  nominal  case.  In  this 
manner,  appropriate  designs  can  be  established  to  meet  safety  margins  with 
more  confidence,  and  will  avoid  costly  redesign  and  retrofitting  at 
downstream  stages  where  added  costs  can  occur  and  where  down  times  are 
difficult  to  tolerate.  The  concept  is  further  illustrated  in  the  sketch  of  Fig  (4). 
Here  the  strain  response  curve  of  the  nominal  case  and  that  of  the  scaled 
version  are  combined  with  strain  to  failure  data  (  coupon  tests)  to  show  fatigue 
results.  Note  the  strain  response  for  the  nominal  case  at  the  highest  dB  level 
available  gives  the  fatigue  value  at  point  A,  while  the  extrapolated  data  for 
this  curve  gives  point  B.  The  scaled  model  being  more  responsive  gives  the 
point  C,  and  when  rescaled  gives  the  point  D  which  differs  slightly  from  the 
extrapolated  point  B  as  it  most  likely  will,  realistically.  More  faith  should  be 
placed  on  data  from  an  actual  fatigue  point  than  a  point  based  on  the  projected 
strain  response  curve.  Note,  Fig  (5)  illustrates  the  winning  virtue  of  the 
scaleable  design.  The  figure  shows  a  hypothetical  set  of  test  data  for  the  (s  ,N) 
for  a  structure  for  various  SPLs  for  the  nominal  case,  open  circles,  and  for  the 
fatigue  results  of  the  scaled  model  ,  closed  squares.  The  scaled  model  was 
assumed  to  be  thinner  here  for  example,  and  that  the  scaled  data  is  also 


957 


rescaled  to  fit  the  nominal  curve  here.  The  most  interesting  aspect  is  shown 
by  the  two  clusters  of  data ,  denoted  as  A  and  B  where  there  are  rough  circles 
about  drawn  about  the  clusters.  Here  the  emphasis  is  that  tests  of  the  scaled 
model  (  and  rescaled  data )  are  used  to  find  the  higher  strain  conditions  which 
cannot  be  found  from  the  nominal  case.  In  both  cases  at  the  highest  strain 
levels,  the  facility  is  used  to  its  limits,  but  with  enough  testing  with  the  thinner 
case,  adequate  data  is  available  to  make  the  prediction  more  accurate  using 
Eq  (2)  for  the  final  correlation  as  shown  here.  The  statistical  scattering  of  the 
scaled  data  will  be  an  accurate  measure  for  the  nominal  case,  particularly 
when  compared  to  estimates  based  on  extrapolation  of  the  strain  response  for 
the  nominal  case.  There  are  many  cautions  to  be  noted  with  this  approach  as 
there  are  with  all  acoustic  fatigue  methods,  and  of  course,  tests.  First,  the 
linearity  of  the  modes,  either  in  unimodal  sine  excitation,  multi-mode  sine, 
narrow  band  or  broadband  random  must  be  carefully  handled.  The  strain 
response  of  individual  locations  throughout  the  structure  must  be  carefully 
monitored  in  calculations  and  tests  so  that  strain  response  is  truly  understood 
and  used  to  define  fatigue  life  carefully.  This  is  difficult  to  do  in  many 
applications  where  widely  varying  conditions  and  durations  require  some  type 
of  Miner  Rule  combination  to  provide  a  true  measure  of  fatigue.  Similarly, 
strain  risers  at  fasteners,  discontinuities,  holes,  frames,  stiffeners,  material 
changes  along  with  temperature  gradients,  temperature  transients,  require 
final  “tweeks”  to  predictions,  regardless.  Nonlinearity ,  especially  in  the 
multi  -mode  case,  is  one  of  the  most  formidable  foes  to  conquer  for  any 
application. 


APPLICATIONS  AND  EXAMPLES 

The  tests  of  an  Aluminum  panel  of  size  10x20  in.  and  with  a  thickness  of 
0.063  thickness  ,  Ref  (19),  will  be  used  to  illustrate  the  technique.  The  panel 
has  approximately  fixed-fixed  edge  conditions,  and  is  quite  nonlinear  in 
terms  of  strain  response.  Ref  (19).  The  measured  strain  response  for  the  panel 
is  shown  in  Fig  (6)  along  with  an  estimated  response  curve  for  a  thinner  panel 
(  0.040  in  )  based  on  the  test  data.  For  this  case  it  is  assumed  that  data  were 
needed  at  175  dB,  while  the  facility  could  only  achieve  164  dB.  The  strain 
response  for  the  thinner  case  was  estimated  using  the  classical  equation  for 
the  amplitude  of  response,  5  ; 


\\fdxdy\pSD,{f)] 


(3) 


where  M  is  the  generalized  mass,  co  is  the  natural  frequency,  §  is  the  viscous 
damping  factor  ,  (j)  is  mode  shape,  PSDp  is  the  pressure  Power  Spectral 
Density,  and  x,y  are  the  positional  coordinates  along  the  plate.  Since  strain  , 
s,  is  proportional  to  the  amplitude. 


958 


s=  (t/2)(3(ti^/ax^)S 


(4) 

Combining  Eq  (3  and  4)  shows  that  the  strain  response  curves  are 
proportional  to  the  thickness  factor,  as  given  by: 

s  2  ~  (  ti  /  t2)  (5) 

However,  it  must  be  noted  that  this  case  is  nonlinear,  and  thus,  this  result  is 
not  exactly  correct,  but  simply  used  for  an  illustration  here.  The  actual  data  for 
the  0.063  thickness  is  extrapolated  to  a  required  175  dB,  showing  a  strain  of 
1000  micro  in/in.  The  estimated  curve  for  the  thickness  of  0.040  in.  shows  ,of 
course,  a  greater  response  at  all  dB  levels  as  it  should,  and  moreover  shows 
that  only  150  dB  are  needed  to  achieve  the  1000  micro  -  strain  condition. 
Moreover,  the  thinner  panel  will  exhibit  large  enough  strains  at  the  lower 
SPLs  to  improve  the  fatigue  curve  where  the  thicker  panel  is  insensitive. 
Taking  the  example  a  step  further,  the  fatigue  point  of  the  nominal  case  is 
shov^Ti  on  a  strain  to  failure  plot  in  Fig  (7) ,  employing  beam  coupon  tests  of 
Ref  (11),  which  were  shown  to  be  excellent  correlators  with  panel  fatigue  in 
the  collection  of  work  in  Ref  (6-13).  The  fatigue  point  for  the  164  dB 
excitation,  800  micro-strain,  is  shown  as  a  triangle,  while  the  extrapolated 
data  for  175  dB  is  shown  as  the  flagged  triangle.  One  test  point  exists  for  the 
nominal  case.  Ref  (20) ,  and  is  shown  by  a  star  symbol.  Data  for  the  thinner 
panel  are  shown  as  circles  at  the  various  strain  to  cycle  count  cases  for  the 
various  SPLs  corresponding  to  the  beam  curve.  Notably,  these  points  can  be 
seen  to  produce  shorter  fatigue  cases  as  they  should  due  to  increased  strains, 
but  note  that  they  are  also  at  lower  frequencies  which  would  give  a  longer  test 
time  than  if  they  were  the  nominal  thickness.  The  scaled  model  is  seen  to 
produce  the  same  point  as  the  extrapolated  case  in  this  hypothetical  case  for 
the  1000  microstrain  case  (  again,  at  two  different  dB  levels  for  two 
thicknesses).  A  SPL  of  150  dB,  rather  benign,  is  seen  to  be  quite  effective. 
The  actual  fatigue  point  at  164  dB  for  the  nominal  case  required  3  hours  and 
was  predicted  to  be  2.8  hrs.  The  estimated  fatigue  for  the  extrapolated  case  of 
175  dB  was  estimated  to  be  1.7  hrs,  while  the  scaled  point  from  the  thinner 
panel  was  estimated  to  be  2.2  hours  which  is  slightly  off,  but  the  Authors 
have  had  to  rely  on  log  plots  for  much  of  the  data  and  thus  lack  someaccuracy. 
Because  of  lack  of  actual  data ,  the  scatter  from  the  estimate  vs  the  test  of  the 
nominal  case  was  used  to  scatter  the  estimate  for  the  1000  microstrain  case, 
flagged  dark  circle,  as  if  the  use  of  Eq  (2)  had  been  employed  directly.  One 
must  be  careful  here,  because  there  can  be  a  vast  difference  between  theory 
and  test,  and  this  can  mislead  inexperienced  persons  applying  these  methods. 
As  noted  earlier,  related  work  in  fluid-structure  and  buffet  ,  actually 
demonstrate  this  type  of  scaling.  To  illustrate,  several  figures  are  republished 


959 


here  to  make  this  point  rather  clear.  Fig  (8)  of  Ref  (8  )  shows  the  dynamic 
bending  strains  in  the  bottom  panel  of  an  otherwise  rigid  fuel  tank  which  is 
being  excited  vertically  with  moving  base  input.  The  vertical  axis  is  strain 
while  the  abscissa  is  the  number  of  g’s  input.  Three  panel  thicknesses  and  four 
depths  of  fluid  (water  in  this  case)  were  used.  Note  the  sharp  nonlinear  effect 
in  the  response,  rather  than  linear  response  growth  as  force  increases. 
Interestingly,  the  data  was  nondimensionalized  into  the  curve  of  Fig  (9),  Ref 
(8),  which  was  originally  intended  for  a  design  chart  to  aid  in  developing 
strain  response  characteristics  for  use  in  fatigue.  This  curve  displays  a 
parameter  of  response  as  the  ordinate  vs  an  excitation  parameter  on  the 
abscissa.  Here,  E  is  Young’s  modulus,  p  is  density,  t  is  thickness,  a  is  the 
panel  length  of  the  short  side,  h  is  fluid  depth,  and  the  subscripts,  p  and  F  refer 
to  panel  and  fluid,  respectively.  A  point  not  realized  previously  is  that  the 
scaling  shows  that  the  thinner  case  can  be  used  to  represent  the  thicker  panel 
under  the  appropriate  conditions  and  when  nonlinearity  is  carefully 
considered.  More  data  with  the  thinner  panels  at  the  extreme  conditions  were 
unfortunately  not  taken  in  several  cases  of  strain  response  because  of  concern 
with  accumulating  too  many  cycles  before  running  the  actual  fatigue  tests; 
else  the  thinner  cases  could  have  shown  even  more  dramatically  the  scale 
effect. 

Buffet  has  been  of  more  interest  in  the  past  15  years  because  of  high  angle  of 
attack  operation  of  several  modem  USAF  fighters.  Much  effort  was  placed 
upon  research  with  accurately  scaled  models  to  detennine  if  these  could  be 
employed  as  in  prior  flutter  work.  The  answer  was  YES!  Several  figures  were 
taken  from  Ref  (16)  to  illustrate  scaling  of  data  from  a  model  of ,  a  fraction 
of  the  size  of  a  fighter,  to  the  full  scale  quantity.  Fig  (10)  shows  the  correlation 
between  scaled-up  model  data,  flight  test,  and  two  sets  of  calculations  over  a 
wide  range  of  aircraft  angle  of  attack  for  the  F/A-18  stabilator.  The  data  is  for 
inboard  bending  and  torsion  moment  coefficients  produced  by  buffeting 
loads.  The  scaled  model  data  correlates  well,  the  calculations  using  Doublet 
Lattice  (DLM)  aerodynamics  is  close,  while  the  strip  theory  is  not  as  accurate. 
Fig  (11)  shows  similar  type  of  data  for  the  F/A-18  Vertical  Tail  for  outboard 
bending  moment  coefficients.  Here  a  wider  range  of  angle  of  attack  was 
considered,  and  again  scaled  model  data  and  calculations  are  close  to  aircraft 
values.  Both  cases  suggest  that  model  data  can  be  used  to  supplement  full 
scale  work  and  that  when  combined  with  theory  ,  are  a  powerful  aid  to  full 
scale  analysis  and  tests.  These  tests  can  be  used  early  in  the  aircraft  design 
cycle  to  insure  full  scale  success. 

CONCLUSIONS  AND  RECOMMENDATIONS 
An  attempt  was  made  to  employ  a  view  of  acoustical  scaling  different  from 
that  usually  taken.  The  idea  is  to  develop  data  for  a  model  that  fits  within  a 
test  facility’s  capability  and  then  by  using  analytical  methods,  adjust  these 
results  to  the  nominal  case  using  factors  from  the  test  based  on  the  ratio  of 


960 


experimental  to  calculated  data.  This  is  analogous  to  the  flutter  model 
approach.  One  example  is  offered,  and  similar  results  from  related  scaling  in 
fluid  -structure  and  buffet  work  were  shown  to  further  the  point.  While  more 
work  is  needed  to  fully  display  the  concept,  enough  has  been  done  to  inspire 
others  to  dig-in  and  more  fully  evaluate  the  approach.  The  Writers  intend  to 
do  more  research,  since  they  fully  appreciate  this  difficult  task. 

REFERENCES 

1.  Scanlon  ,  R.H.,  and  Rosenbaum,  R.,  “  Introduction  to  The  Study  of  Aircraft 
Vibration  and  Flutter”,  The  MacMillian  Company,  New  York,  1951 

2.  Bisplinghoff,  R.L.,  Ashley  ,  H.  and  ,  Halfman,  R.  L.,  “Aeroelasticity”, 
Addision-Wesley  Publishing  Co.,  NY,  Nov.  1955,  pp.  695-787 

3.  Ferman,  M.  A.,  “Conceptual  Flutter  Analysis  Techniques  -  Final  Report” 
Navy  BuWeps  Contract  NO  w  64-0298-c,  McDonnell  Report  F322,  10  Feb. 
1967 

4.  Ferman,  M.A.  and  Unger,  W.  H.,  “Fluid-Structure  Interaction  Dynamics  in 
Fuel  Cells”,  17th  Aerospace  Sciences  Meeting,  New  Orleans,  La.  Jan  1979 

5.  Ferman,  M.A.  and  Unger,  W.  H.  ,  “Fluid-Structure  Interaction  Dynamics  in 
Aircraft  Fuel  Cells”,  AlAA  Journal  of  Aircraft,  Dec.  1979 

6.  Ferman,  M.A.  ,  et  al  ,  “  Fuel  Tank  Durability  with  Fluid-Structure 
Interaction  Dynamics  ,”  USAF  AFWAL  TR-83-3066,  Sept.  1982 

7.  Ferman,  M..A.,  Unger,  W.  H.,  Saff,  C.R.,  and  Richardson,  M.D.,  “A  New 
Approach  to  Durability  Predictions  For  Fuel  Tank  Skins”  ,  26th  SDM, 
Orlando,  FL,  15-17  April  1985 

8.  Ferman,  M.  A.  ,  Unger,  W.  H.,  Saff,  C.R.,  and  Richardson,  M.D.  ,  “  A 
New  Approach  to  Durability  Prediction  For  Fuel  Tank  Skins”,  Journal  of 
Aircraft,  Vol  23,  No.  5,  May  1986 

9.  Saff,  C.R.,  and  Ferman,  M.A,  “Fatigue  Life  Analysis  of  Fuel  Tank  Skins 
Under  Combined  Loads”,  ASTM  Symposium  of  Fracture  Mechanics  , 
Charleston,  SC,  21  March  1985 

10.  Ferman,  M.A.,  Healey,  M.D.,  Unger,  W.H.,  and  Richardson,  M.D., 
“Durability  Prediction  of  Parallel  Fuel  Tank  Skins  with  Fluid-Structure 
Interaction  Dynamics”,  27th  SDM,  San  Antonio,  TX,  19-21  May  1986 

11.  Ferman,  M.A,  and  Healey,  M.D.,  “Analysis  of  Fuel  Tank  Dynamics  for 
Complex  Configurations,  AFWAL  TR  -87-3066,  Wright-Patterson  AFB,  OH, 
Nov  1987 

12.  Ferman,  M.A.,  Healey,  M.D.  and  Richardson,  M.D.,”  Durability  Prediction 
of  Complex  Panels  With  Fluid-Structure  Interaction”,  29th  SDM, 
Williamsburg,  VA,  18-20  April  1988 

13.  Ferman,  M.A.,  Healey,  M.D.,  and  Richardson,  M.D.,  “A  Dynamicisf  s 
View  of  Fuel  Tank  Skin  Durability,  AGARD/NATO  68th  SMP,  Ottawa, 
Canada,  23-28  April  1989 

14.  Zimmerman,  N.H.  and  Ferman,  M.A.,  “Prediction  of  Tail  Buffet  Loads 
for  Design  Applications,  USN  Report,  NADC  88043-30,  July  1987 


961 


15.  Zimmerman,  N.H.,  Ferman,  M.A.,  Yurkovich,  R.N,  “Prediction  of  Tail 
Buffet  Loads  For  Design  Applications”,  30th  SDM,  Mobil,  AL  ,  3-5  April 
1989 

16.  Ferman,  M.A.,  Patel,  S.R.,  Zimmemian,  N.H.,  and  Gerstemkom,  G.,  “  A 
Unified  Approach  To  Buffet  Response  Response  of  Fighters”, AGARD/NATO 
70th  SMP,  Sorrento,  Italy,  2-4  April  1990 

17.  Cote,  M.J.  et  al,  “Structural  Design  for  Acoustic  Fatigue”,  USAF  ASD- 
TR-63-820,  Oct  1963 

18.  Rudder,  F.F.,  and  Plumblee,  H.E.,  Sonic  Fatigue  Guide  for  Military 
Aircraft”  AFFDL-TR-74-1 12,  Wright-Patterson  AFB,  OH,  May  1975 

19.  Jacobs,  J.H.,  and  Ferman,  M.A.  ,  Acoustic  Fatigue  Characteristics  of 
Advanced  Materials  and  Structures,  “  AGARD/NATO  SMP  ,  Lillehammer, 
Norway,  4-6  May  1994 

20.  McDonnell  Douglas  Lab  Report,  Tech.  Memo  253.4415,  Acoustic  Fatigue 
Tests  of  Four  Aluminum  Panels,  Two  With  Polyurethene  Sprayon”,  27  June 
1984 


Figure  1  -  Standard  Acoustic  Fatigue  Design  Method 


962 


MICROSTRAIN  - 


HYPOTHETICAL  EXAMPLE 


SOUND  PRESSURE  LEVEL  -dB  N- CYCLES  TO  FAILURE 


Figure  2  -  General  Method  for  Estimating  Fatigue  Life  at  SPL 
above  Test  Facility  Capability 


Figure  3  -  Scaling  Method  Fits-in  with  General  Design 
Cycle  for  Acoustic  Fatigue 


963 


SOUND  PRESSURE  LEVEL  -  dB 


N@  cIBreq  ,  Nom.  Ext. 
N  @  dBj^ ,  Re-scaled 
N  @  dBj^ ,  Scaled  Model 
N  -  CYCLES  TO  FAILURE 


Figure  4  -  Hypothetical  Examples  of  Acoustic  Scaling 
to  Tests  at  Higher  SPLs 


HYPOTHETICAL  CASE 


Figure  5  -  Statistical  Aspects  of  Scaling 


MICROSTRAIN  - 


^  MDC  Tests 
Ref.  (19) 


Figure  6  -  Strain  Response  of  an  Aluminum  Panel 

(10  X  20  X  0.063  in.  7075  T6)  Narrowband  Random 


Figure  7  -  Example  of  scaled  Model  of  the  10  x  20  in. 
Aluminum  Panel 


965 


Symbol  Panel  Fluid 
Thickness  Depth 

(in.)  (in.) 

□ 

0.032 

11.0 

0 

0.040 

11.0 

o 

0.063 

11.0 

0 

0.032 

8.0 

A 

0.040 

8.0 

o 

0.063 

8.0 

0 

0.032 

4.0 

0.040 

4.0 

• 

0.063 

4.0 

Figure  8  -  Dynamic  Strain  vs  Excitation  Level  -  Sine 


Figure  9  -  Dynamic  Strain  Parameter  vs  Input  Parameter 

Mean  Strain  at  Fluid  Depth 


966 


0.04 


Inboard  Bonding 


Inboard  Torsion 


0.032 

RMS  Buffet  0.024. 

Moment 
Coeffidant, 

0.01  e 

0.008 

0  . 

0  4  8  12  16  20  24 

Angle  of  Attack  -  degrees 


A 

C 

1 

1 

r 

1  0 

< 

) 

• 

0 

_ ^ 

— 

Right  t» 
Scaladt 
Calculal 
italculat 

— 

St 

ipmods 

0d(str^ 

ed(OUl 

Itast 

theory) 

) 

— 

A 

L 

i 

> 

(j 

I  o 

( 

0 

) 

0  4  8  12  16  20  24 


Angie  of  Attack  -  degrees 


Figure  10  -  F/A-18  Stabilator  Buffet  Correlation  Study 


0.048 

0.04 

0.032 

RMS 
Moment 

CoefficiantO.024 
BM/(QL3) 

0.016 

0.008. 

0 

16  24  32  40  48  56 

Angle  of  Attack  -  degrees 

Figure  11  -  F- 18  Vertical  Tail  Buffet  Response  Moment  Coefficients  for 
Angle-of-attack  Variations 


Outboard  Bending  Moment  (70%  Span,  45%  Chord) 

j— I  j  j 


— 

□  Ca 
OMC 
A  FI 

— 

Liculatsd 

ideiscaX 

gniTastf 

— 

d4ip(MS 

517 

— 

WTTeet 

— 

51.  Run 

— 

56) 

( 

A 

1  ( 

'  *, 

)  < 

) 

" 

J 

h 

*  ( 

> 

( 

I 

1 

t 

1 

Zi 

1 

967 


968 


ACOUSTIC  FATIGUE  II 


THE  DEVELOPMENT  AND  EVALUATION  OF  A  NEW 
MULTIMODAL  ACOUSTIC  FATIGUE  DAMAGE  MODEL 


Howard  R  Wolfe 
WL/FIBG  Bidg24C 
2145  Fifth  St  Ste2 
Astronautics 

Wright-Patterson  AFB,  OH 
45433-7006 
USA 


Robert  G.  White 
Head  of  Department 
Department  of  Aeronautics  and 

University  of  Southampton 
Southampton,  S017 IBJ 
UK 


ABSTRACT 

A  multimodal  fatigue  model  has  been  developed  for  flat  beams  and  plates. 
The  model  was  compared  with  experimental  bending  resonant  fatigue 
lifetimes  under  random  loading.  The  method  was  accurate  in  predicting 
cantilevered  beam  fatigue  lifetimes,  but  under  predicted  clamped-  clamped 
beam  test  results.  For  the  clamped  plate  tests,  one  calculation  was  accurate 
and  the  other  predicted  about  half  the  test  lifetimes.  The  comparisons  and 
the  parameters  affecting  them  are  presented. 

INTRODUCTION 

While  the  single  mode  acoustic  fatigue  theory  is  satisfactory  for  sound 
pressure  levels  around  158  dB  overall  and  below,  there  is  evidence  in  the 
literature  [1,  2,  and  3]  that  above  this  level  the  accuracy  of  the  simple 
response  prediction  method  decreases  with  increasing  sound  pressure 
levels.  The  purpose  of  the  paper  is  two  fold,  first  to  develop  a  multimodal 
acoustic  fatigue  life  prediction  model  ^d  secondly,  to  evaluate  its  accuracy 
in  estimating  the  fatigue  life  theoretically  by  comparing  predictions  with 
experimental  results. 

FATIGUE  MODEL  DEVELOPMENT 

Many  fatigue  models  are  found  in  the  literature.  The  Miner  single  mode 
model  used  by  Bennouna  and  White  [4]  and  Rudder  and  Plumblee  [5]  was 
selected  to  develop  a  multimodal  nonlinear  model.  The  fundamental 
formulation  is  given  by, 

N.-fs^r  «) 


969 


where  N,.  is  the  total  number  of  cycles  to  failure,  P(e)  is  the  peak  strain 
probability  density.  N  is  the  total  number  of  cycles  to  failure  at  incremental 
constant  amplitude  strain  levels  derived  from  a  sinusoidal  strain  versus 
cycles  to  failure  curve.  To  calculate  the  fatigue  life  in  hour,  Eq  (1)  can  be 
expressed  as, 

t  (hours) = 


where  t  is  time,  Pp  (sd)  is  the  peak  standard  deviation  probability  density, 
Nc  is  the  total  number  of  cycles  to  failure  at  a  specified  strain  level  and  ^  is 
the  cyclic  frequency.  When  the  mean  value  is  not  zero,  which  is  the  case 
with  axial  strain  in  the  beam  or  plate,  the  rms  value  is  not  the  standard 
deviation.  The  standard  deviation  is  usually  employed  to  compute  the  time 
to  failure.  Most  of  the  S-N  curves  or  e  -N  curves  are  approximated  as  a 
straight  line  on  a  logarithmic  graph.  The  relationship  between  the  surface 
strain  and  the  cycles  to  failure  is  then. 


8  = 


(3) 


where  K  is  a  constant  and  a  the  slope  of  a  straight  line  on  a  log-log  graph. 
The  cyclic  frequency  4  for  a  single  mode  case  is  taken  as  the  frequency  of 
the  associated  resonance.  Two  types  of  peak  probability  density  techniques 
were  investigated  from  multimodal  nonlinear  strain  responses  [3].  These 
were  called  major  peaks  and  minor  peaks.  The  major  peaks  were  counted 
for  the  largest  peaks  between  zero  crossings.  The  minor  peaks  were 
counted  for  all  stress  reversals  or  a  positive  slope  in  the  time  history 
followed  by  any  negative  slope.  The  effective  cyclic  frequency  is  much 
lower  for  the  major  peak  count  than  the  minor  peak  count.  However,  the 
peak  probability  density  functions  or  PPDFs  compared  for  these  two  cases 
were  almost  the  same.  The  major  peak  method  was  selected  for  further 
study. 

Given  a  particular  peak  probability  density  curve  from  a  measured 
response  in  an  experiment,  the  number  of  peaks  and  the  sampling  time  tg 
can  be  used  to  determine  the  effective  multimodal  cyclic  frequency, 

fcm  =  number  of  peaks  /  tg  (4) 


970 


where  is  the  effective  multimodal  cyclic  frequency.  Substituting  the 
multimodal  cyclic  frequency  into  Eq  (4), 


t  (hours)  = 


Pp(sd) 

K/(e)‘/“ 


3600xfcm 


(5) 


This  model  accounts  for  the  effects  of  axial  strains  which  cause  the  mean 
value  not  to  be  zero,  nonlinear  response  and  multimodal  effects.  If  the 
mean  value  is  zero,  then  the  standard  deviation  is  equal  to  the  rms  value 
and  Eq  (5)  reduces  to  Eq  (2). 

FATIGUE  MODEL  COMPARISON  WITH  EXPERIMENTAL 

RESULTS 


The  peak  probability  density  function  (PPDF)  is  needed  or  preferably 
the  time  history  from  strain  or  dynamic  response  measurements  to 
evaluate  the  fatigue  model  developed.  Also  needed  are  sinusoidal  8  -N 
curves  for  the  structure,  and  knowledge  of  its  boundary  conditions  and  the 
equivalent  multimodal  cyclic  frequency. 

Two  types  of  peak  probability  density  techniques  were  investigated  from 
multimodal  nonlinear  strain  responses  [3].  These  were  called  major  peaks 
and  minor  peaks.  The  major  peaks  were  counted  for  the  largest  peaks 
between  zero  crossings.  The  nunor  peaks  were  counted  for  all  stress 
reversals  or  a  positive  slope  in  the  time  history  followed  by  any  negative 
slope.  The  effective  cyclic  frequency  is  much  lower  for  the  major  peak  coimt 
than  the  nunor  peak  coimt  However,  the  peak  probability  density 
functions  or  PPDFs  compared  for  these  two  cases  were  almost  the  same. 
The  major  peak  method  was  selected  for  further  study. 

Comparison  with  Beam  Data: 

The  K  and  a  terms  were  calculated  from  8-N  and  S-N  data,  where  S 
is  stress,  using  Eq  (3).  Selecting  two  values  of  strain  and  their 
corresponding  cycles  to  failure,  yields  two  simultaneous  equations 
which  were  solved  for  K  and  a.  Two  sinusoidal  E-N  curves  for 
BS1470-NS3  aluminum  alloy  which  has  a  relatively  low  tensile  strength 
were  obtained  from  Bennouna  and  White  [4  Fig  8].  These  were  for  a 
cantilevered  beam  and  a  clamped-clamped  (C-C)  beam  as  shown  in  Fig 
1.  The  K  and  a  terms  calculated  were  used  to  compute  the  cycles  to 
failure,  Nc,  for  each  strain  level.  Table  I  shown  is  the  same  as  Table  I  in 
reference  [4]  except  was  calculated  from  Eq  (3)  to  sum  the  damages. 


971 


Delta  is  the  sample  size.  The  cyclic  frequency  was  for  one  mode  the  one 
resonant  response  frequency.  The  time  to  failure  in  hours  using  Eq  (2) 
for  the  cantilevered  beam  was  16.6  hours  compared  with  16.2  predicted 
theoretically  [4],  15.3  and  15.9  obtained  experimentally  [4].  Both 
theoretical  results  were  essentially  equal,  but  slightly  higher  than  the 
test  results.  The  time  to  failure  from  Eq  (2)  for  the  (C-C)  beam  was  3.04 
hours  compared  with  2.53  theoretically  [4]  and  5.25  and  5.92 
experimentdly  [4].  Both  theoretical  results  were  about  one  half  of  the 
test  results.  The  K  and  a  terms,  the  theoretical  fatigue  life  times  and  the 
experimental  fatigue  life  times  are  listed  in  Table  H.  The  table  contains 
three  sections:  lifetimes  calculated  using  a  strain  gauge  PPDF,  a 
displacement  PPDF  and  the  Gaussian  and  Rayleigh  PDFs.  As  noted  in 
reference  [4],  failure  occurred  much  earlier  for  the  C-C  beam  than  the 
cantilevered  beam  for  the  same  strain  level.  This  was  attributed  to  the 
influence  of  a  large  axial  strain  in  the  clamped-  clamped  beam. 

Comparison  with  Plate  Data: 

Two  fatigue  tests  were  conducted  to  provide  some  additional  limited 
data  for  comparison  with  the  fatigue  model  developed.  These  tests 
used  the  base  excitation  method  with  a  1.09x10®  N  (20,000  Ibf) 
electrod)mamic  shaker.  The  clamping  fixture  consisted  of  a  flat 
aluminum  alloy  6061-T6  plate  19  mm  thick  and  four  clamping  bars  of 
equal  thickness.  The  radius  of  curvature  of  the  clamping  edges  was  4.76 
mm  to  prevent  early  fatigue  failure.  A  four  bar  clamping  arrangement 
was  selected  to  prevent  buckling  of  the  plate  while  torquing  the 
clamping  bolts.  The  undamped  size  was  254  x  203  x  1.30  mm  which 
results  in  a  1.25  aspect  ratio.  Strain  gauges  were  bonded  along  the  center  of 
the  larger  dimension  (SG  2)  and  at  the  center  of  the  plate  (SG  3). 
Displacements  were  measured  with  a  scanning  laser  vibrometer  at  the 
center  of  the  plate.  An  accelerometer  was  moimted  on  the  shaker  head  to 
determine  the  acceleration  imparted  to  the  damped  plate.  A  flat 
acceleration  spectral  density  was  used  between  100-1500  Hz.  Recordings 
were  taken  at  increasing  levels  of  exdtation  up  to  the  fatigue  test  level.  The 
time  to  detecting  the  first  fatigue  crack  was  recorded  for  each  plate. 

The  constants  K  and  a  were  calculated  from  random  single  mode  S-N 
data  for  7075-T6  aluminum  alloy  [5  p  489]  shown  in  Fig  1,  with 

K  =  1.01x10^^  and  a  =  -  0.175.  The  rms  stress  ( Srms )  ^^s  changed  to  rms 
strain,  Srms  =  EBrms,  where  E  is  Young's  modulus.  The  stress  was 
measured  half  way  between  two  rivets  along  the  center  line  between  the 
rivets  on  the  test  specimen.  The  strain  gauge  location,  stress 
concentrations,  and  the  boundary  conditions  greatly  affect  the  strain 


972 


level  measured.  Correction  factors  are  needed  for  a  different  set  of 
conditions  and  to  convert  random  data  to  sinusoidal  data.  Sinusoidal 
£  -N  bending  coupon  curves  for  7075-T6  aluminum  alloy  were  difficult  to 
find.  S-N  curves  were  found  for  an  aerospace  material  with  both 
sinusoidal  and  random  excitations.  These  curves  were  nearly  parallel. 
The  sinusoidal  strain  was  1.38  times  larger  than  the  random  strain  for 

10^  cycles.  Multiplying  the  constant  K  for  the  7075-T6  material  by  1.38 
resulted  in  K  =  1.40x10^^ . 

Early  strain  gauge  failures  prevented  strain  measurement  above  500 
microstrain  with  20.7  g  rms  shaker  excitation.  The  fatigue  test  level  was 
115  g  rms  and  the  response  contained  at  least  six  frequency  response 
peaks.  The  major  peak  strain  PPDFs  were  determined  for  5.32  g  rms 
and  20.7  g  rms  as  shown  in  Fig  2  with  the  Gaussian  PDF.  Compared 
with  the  Gaussian  distribution,  an  increased  number  of  peaks  occurred 
greater  than  1  sigma  and  smaller  than  -1  sigma.  Also  a  larger  number  of 
peaks  occurred  around  zero.  The  PPDF  determined  from  the  20.7g  rms 
test  case  was  used  to  predict  fatigue  life,  but  a  new  strain  estimate  was 
needed  since  the  excitation  level  increased  5.6  times.  The  displacement 
is  directly  related  to  the  strain  at  each  excitation  level.  Displacement 
measurements  at  the  fatigue  test  level  were  used  to  estimate  the  strain 
level  shown  in  Fig  3.  The  estimated  strain  from  the  figure  was  770 
microstrain  for  SG  2.  The  scale  of  the  displacement  measurements  was 
adjusted  to  coincide  with  strain  measurements  at  increasing  increments 
of  shaker  excitation. 


The  equivalent  cyclic  multimodal  frequency  is  needed  to  predict  the 
fatigue  life.  Prediction  of  the  linear  modal  frequencies  is  carried  out  by 
a  variety  of  methods.  Usually  the  first  mode  prediction  is  the  most 
accurate.  The  cyclic  multimodal  nonlinear  frequencies  have  been 
studied  for  two  clamped  beams  and  two  clamped  plates  [3].  These  were 
based  upon  the  peak  probability  density  functions  (PPDFs)  where  the 
peaks  were  counted  for  a  specific  time  interval,  from  which  the 
nonlinear  cyclic  multimodal  frequencies  were  calculated.  Generally  the 
resonant  frequencies  increased  with  increasing  excitation  levels.  Those 
for  the  two  beams  increased  more  rapidly  than  those  for  the  two  plates. 
Very  little  change  was  noted  for  the  plates.  The  equivalent  cyclic 
miiltimodal  frequency  determined  via  Eq  (4)  from  the  SG  2  PPDF  was 
348  Hz. 

The  time  to  failure  in  hours  predicted  using  Eq  (5)  and  the  parameter 
mentioned  above  for  the  clamped  plate  was  0.706  hours  compared  with 
1.17  and  0.92  shown  in  Table  11.  The  predicted  result  was  slightly  lower 


973 


than  the  test  results.  A  Srms’^  curve  was  calculated  with  the  test 
lifetimes  available,  by  determining  a  new  constant  K,  assuming  the 
slope  was  the  same  as  for  the  riveted  coupon  and  applying  the 
sinusoidal  correction  factor.  The  time  to  failure  in  hours  predicted 
using  Eq  5  and  the  calculated  e-N  curve  for  the  clamped  plate  was 
0.274  hours  compared  with  1.17  and  0.92  shown  in  Table  E.  The  cyclic 
multimodal  frequency  used  was  the  same  at  that  determined  from  SG  2 
and  the  same  strain  was  used.  This  prediction  was  about  1/3  of  the  test 
results.  This  method  incorporates  the  failure  data  at  two  points. 

The  displacement  PPDF  shown  in  Fig  4  was  used  to  predict  fatigue 
life.  The  number  of  displacement  peaks  increased  significantly  above 
the  strain  PPDF  around  1  sigma  and  -1  sigma.  The  large  number  of 
peaks  around  zero  was  similar  to  the  characteristics  observed  in  the 
strain  PPDF.  The  equivalent  cyclic  multimodal  frequency  was  375  Hz, 
slightly  higher  than  that  determined  from  the  strain  PPDF.  However, 
the  same  frequency  (348  Hz)  was  used  to  predict  fatigue  life.  The  time  to 
failure  in  hours  using  Eq  (5)  for  the  clamped  plate  was  1.15  hours  with 
the  riveted  e-N  curve  and  0.446  with  the  calculated  8-N  curve  shown 
in  Table  E.  This  PPDF  improved  considerably  the  prediction.  The 
Gaussian  and  Rayleigh  PDFs  were  used  to  predict  the  time  to  failure 
with  the  same  parameters  as  those  used  with  the  riveted  e-N  curve. 
The  lifetime  using  the  displacement  PPDF  was  1.15  hours,  using  the 
Gaussian  PDF,  0.600  hours  and  using  the  Rayleigh  PDF,  0.237  hours,  as 
shown  in  Table  E.  The  Gaussian  PDF  under  predicts  by  a  factor  of  2. 
The  Rayleigh  PDF  under  predicts  by  a  factor  of  5. 

The  spread  sheets  containing  PPDF  /  Ncm  data  for  various  sigma 
values  were  used  to  determine  damage  accumulation  shown  in  Fig  5. 
Almost  55%  of  the  normalized  damage  occurs  between  -2  and  -1  sigma 
and  38%  of  the  damage  between  1  and  2  sigma  using  the  displacement 
PPDF.  However,  the  damage  is  spread  more  evenly  using  the  available 
strain  gauge  PPDF.  The  strain  gauge  PPDF  was  recorded  at  a  much 
lower  level  than  the  displacement  PPDF.  The  damage  accumulation 
compared  more  closely  to  Gaussian  PDF  than  the  Rayleigh  function. 


Damage  Model  with  a  Specific  Function  Describing  the  PPDF: 

A  curve-fitting  routine  was  used  to  determine  a  mathematical  function 
for  a  high  level  strain  gauge  PPDF  for  the  clamped  shaker  plate.  The 
most  important  part  of  the  fit  is  outside  the  range  of  -1  to  1  sigma,  since 
most  of  the  damage  accumulation  occurs  outside  this  range.  The 
highest  ranking  function  was  a  tenth  order  polynomial  followed  by 


974 


ninth  and  eighth  order  polynomial  fits.  The  goodness  of  fit  in  order 
from  1  to  14  ranges  from  0.9775  to  0.9625,  which  are  very  close 
statistically.  The  tenth  order  polynonual  is, 

y  =  a  4*  bx  +  cx^  +  dx^  +  ex"^  +  fx^  +  gx^  +  hx^  +  ix^  +  jx^  +  kx^^  (6) 

where  a=0.346,  b=-0.0148,  c=-0.137,  d=-0.054,  e=0.090,  f=0.043,  g=- 
0.0400,  h=-0.00976,  i=  0.00722,  j=6.85xl0"‘,  k=<4.36xl0'"  .  Ranked 
fourteenth  is  a  natural  logarithmic  function.  The  function  and  its 
coefficients  are, 

Iny  =  a  +  bx  +  cx^  +  dx^  +ex^  4-fx^  (7) 


where  a=-1.088,  b=-0.1191,  c=-0.1.302,  d=0.0104,  e=-0.0653,  and  f=0.0079. 
The  function  fits  the  test  data  similar  to  the  tenth  order  polynomial  and 
may  be  easier  to  use.  Ranked  forty-first  is  a  Gaussian  function.  The 
function  and  its  coefficients  are. 


y  =  a  +  b  exp|o.5[(x  -  c)  /  d]^  | 


(8) 


where  a=-0.0968,  b=0.4485,  c=-0.050  and  d=1.45.  The  function  fits  better 
for  sigma  values  of  2  or  greater  than  those  of  -2  sigma  and  greater.  A 
constant  coefficient  is  used  to  fit  the  Gaussian  function  to  permit  shifting 
the  function  to  fit  the  test  data.  This  equation  can  be  used  in  the  PPDF 
in  Eq  (5), 

a  +  b  exp  [-0.5 [(x  -  c)  /  d]^  } 

-  [K/(e)f“ 


3600£ 


■cm 


(9) 


where  a=-0.0968,  b=0.4485,  c=-0.050  and  d=1.45. 


CONCLUSIONS 

The  prediction  of  multimodal  fatigue  life  is  primarily  dependent  upon 
the  peak  probability  density  function  (PPDF)  which  changes  shape  with 
increasing  excitation  levels.  The  next  in  order  of  importance  is  the 
sinusoidal  e  -N  bending  fatigue  curve  and  finally  the  effective  multimodal 
cydic  frequency. 

A  multimodal  fatigue  model  was  developed  with  the  PPDF  estimated 
from  a  form  of  the  Gaussian  function  being  useful  especially  in  the 


975 


range  of  cydes  to  failure  from  10^-10^.  The  lifetime  predication 
calailations  for  the  damped-clamed  beam  was  about  one  half  the 
experimental  value.  For  the  plate,  the  calculations  was  about  one  half 
the  experimental  value.  Using  riveted  coupon  fatigue  data,  the 
calculation  was  accurate. 


REFERENCES 

1.  B.L.  Clarkson,  April  1994,  '‘Review  of  sonic  fatigue  technolog/', 
NASA  contractor  report  4587,  NASA  Langley  Research  Center, 
Hampton,  Virginia. 

2.  R.G.  White,  October  1978,  "A  comparison  of  some  statistical 
properties  of  the  responses  of  aluminium  alloy  and  CFRP  plates  to 
acoustic  excitation".  Composites  9(4),  125-258. 

3.  H.F.  Wolfe,  October  1995,  "An  experimental  investigation  of 
nonlinear  behaviour  of  beams  and  plates  excited  to  high  levels  of 
dynamic  response",  PhD  Thesis,  University  of  Southampton. 

4.  M.  M.  Bennouna,  and  R.  G.  White,  1984,  "The  effects  of  large 
vibration  amplitudes  on  the  dynamic  strain  response  of  a  clamped- 
clamped  beam  with  consideration  on  fatigue  life".  Journal  of  Sound  and 
Vibration,  96  (3),  281-308. 

5.  J.  R.  Ballentine,  F.  F.  Rudder,  J.  T.  Mathis  and  H.E.  Plumblee, 

1968,  'Refinement  of  sonic  fatigue  structural  design  criteria",  AFFDL  TR 
67-156,  AD831118,  Wright-Patterson  AFB,  Ohio. 


TABLE  I 

FATIGUE  CALCULATIONS  USING  EQUATION  5.9 

CANTILEVERED  BEAM  BS  1470-NS3  (REF  4  FIG  8) 


e/sd 

sd 

P£ 

e 

ll£ 

Nc=(2.172 

xl0“/sd)“““ 

PPDF  X  A 

PPDF  X  A/Nc 

t 

hours 

0.5 

H 

213 

15079045.41 

n 

1.41256E-07 

1 

|g 

425 

4390936.857 

mam 

7.06000E-07 

1.5 

425 

638 

2125446.064 

WBrn 

1.12917E-06 

2 

425 

850 

1273261.536 

0.138 

1.08383E-06 

2.5 

425 

1063 

854026.6252 

0.060 

7.02554E-07 

3 

425 

1275 

617189.6449 

0.018 

2.91645E-07 

3.5 

TOTAL 

425 

1488 

468372.0662 

0.005 

0.984 

1.06753E-07 

4.16121E-06 

16.7 

976 


TABLE  n 

SUMMARY  OF  FATIGUE  CALCULATIONS 


Figure  1  8  -N  curves  for  aluminum  alloys  tested. 


Figure  2  Normalized  strain  PPDF  comparison  with  a  Gaussian  PDF. 


978 


NORMALIZED  DISPLACEMENT 
MOJORPPDF 


2 


Figure  4  Normalized  displacement  PPDF  comparisons, 


9 


980 


ACOUSTIC  FATIGUE  AND  DAMPING  TECHNOLOGY  IN 
COMPOSITE  MATERIALS 


By  B.  Benchekchou  and  R.G.  White 


Abstract 

Considerable  interest  is  being  shown  in  the  use  of  composite  materials  in 
aerospace  structures.  Important  areas  include  development  of  a  stiff,  lightweight 
composite  material  with  a  highly  damped,  high  temperature  polymer  matrix 
material.  The  study  described  in  this  paper  concerns  the  application  of  such 
material  in  the  form  used  in  thin  skin  panels  of  aircraft  and  investigation  of  its 
fatigue  properties  at  room  and  high  temperature.  For  this  purpose,  flexural  fatigue 
tests  have  been  carried  out  at  two  different  temperatures  and  harmonic 
three-dimensional  FE  analyses  were  performed  in  order  to  understand  the 
dynamic  behaviour  of  plates.  Random  acoustic  excitation  tests  using  a 
progressive  wave  tube,  up  to  an  overall  sound  pressure  level  of  162  dB,  at  room 
temperature  and  high  temperatures  were  also  performed  in  order  to  investigate  the 
dynamic  behaviour  of  panels  made  of  the  materials.  Various  methods  for 
including  damping  in  the  structure  were  examined  when  parameter  studies  were 
carried  out,  and  conclusions  have  been  drawn  concerning  optimal  incorporation 
of  a  highly  damped  matrix  material  into  a  high  performance  structure. 

1-Introduction 

Significant  areas  of  primary  and  secondary  structures  in  military  aircraft  operate 
at  high  temperature  and  are  subjected  to  high  levels  of  random  acoustic  loading, 
because  of  their  closeness  to  jet  effluxes.  There  is  then  a  need  to  develop  a  carbon 
fibre  reinforced  plastic  material  with  a  high  temperature  polymer  matrix  and  high 
fatigue  resistance.  Highly  damped  composite  structures  should  be  developed  in 
order  to  better  resist  dynamic  loading  and  to  have  an  enhanced  fatigue  life.  Work 
previously  carried  out  on  improving  the  damping  in  fibre  reinforced  plastic  (FRP) 
composites  as  well  as  the  number  of  approaches  which  can  be  taken  to  improve 
the  damping  properties  of  polymeric  composites  have  been  summarised  in  [1]. 
The  aim  of  the  research  described  here  was  to  study  lightweight  composite 
materials  with  a  highly  damped,  high  temperature  polymer  matrix  material,  by 


981 


investigating  its  mechanical  and  acoustic  fatigue  properties,  the  latter  investigation 
being  carried  out  using  thin,  multilayered  plates. 

2-ExperimentaI  work 

For  this  type  of  study,  two  adequate  prepregs  were  highlighted  after  investigation: 
SE300  and  PMR15.  The  SE300  material  was  carbon  fibre  reinforced  prepreg  of 
(0°/90°)  woven  form,  0.25  mm  thick  and  had  60%  fibre  volume  fraction,  with  no 
suitable  data  available  on  the  material  properties.  Dynamic  mechanical  thermal 
analyser  (DMTA)  analyses  carried  out  on  four  specimens  (  20  nun  long  and  12 
mm  wide)  with  different  lay-ups  i.e.  (0°/90°)4,  (+45°/-45°)4,  (0°/90°,  45°/45°)s 
and  (45®/45°;0®/90°)s,  allowed  to  get  provided  the  material  properties.  Results 
from  DMTA  analyses  are  shown  in  Table  l.a  where  the  loss  factor  and  the 
Young’s  modulus  values  at  40^C  and  at  the  glass  transition  temperature  Tg  are 
presented.  The  loss  factor  values  varied  from  0.0097  to  0.085  for  a  range  of 

temperature  from  40®C  to  300®C. 

The  PMR15  prepreg  was  also  of  (0^/90^)  woven  carbon  form  and  had  58% 
volume  fraction.  Six  DMTA  specimens  having  the  following  lay-ups:  (0®/90®)4, 
(00/90^)8,  (4450/-450)4,  (+450/-45<^)8,  (0^/90^-,  450/450)s  and 

(450/450.00/900)^  were  made.  Results  from  DMTA  analyses  showed  that  the  loss 
factor  values  varied  from  0.0129  to  0.0857  for  a  range  of  temperature  from  40®C 
to  400®C,  with  a  value  of  loss  factor  of  0.1293  at  375®C,  the  maximum 
temperature  for  normal  use  being  352^C.  The  loss  factor  and  Young’s  modulus 
values  at  40®C  and  Tg  are  given  in  Table  l.b. 

Mechanical  behaviour  of  the  selected  materials 

The  fatigue  characteristics  of  these  new  materials  were  investigated  and  results 
were  compared  with  those  of  well  established  structural  materials.  Mechanical 
fatigue  tests  of  SE300  and  PMR15  samples  using  "sinusoidal"  loading  at  a  chosen 
maximum  strain  level,  i.e.  8000  pS  were  carried  out  and  performances  compared 
to  that  of  an  XAS/914  sample.  A  mechanical  (flexural)  fatigue  rig  was  used  for 
this  purpose  to  test  specimens  in  a  cantilevered  configuration.  Details  of  the  rig  are 
available  in  [2].  The  particular  clamp  used  was  designed  by  Drew  [3]  to  induce 
damage  in  the  centre  of  the  specimen  instead  of  having  edge  damage,  i.e.  peeling 
while  flexural  tests  are  carried  out.  In  order  to  investigate  the  performance  of  these 
new  materials  at  high  temperature,  fatigue  tests  were  also  carried  out  on  samples, 

at  210^C  .  This  was  achieved  by  using  a  heating  system  which  consisted  of  two  air 


982 


blowers  (electronically  regulated  hot-air  guns)  positioned  at  40  mm  above  and 
below  the  specimens,  which  allowed  specimens  to  be  tested  at  a  uniform  temperature 

of  210±5°C.  The  aim  of  the  mechanical  fatigue  test  was  to  determine  the  number  of 
loading  cycles  needed  for  damage  to  occur  and  its  subsequent  growth  rate  in 
cyclically  loaded  composite  specimens  of  SE300  and  PMR15  matrix  materials  .  The 
samples  were  140  mm  long,  70  mm  wide  and  2  mm  thick.  Fatigue  tests  of  SE300  (S3 
and  S4)  and  PMR15  (PI  and  P2)  specimens,  at  room  temperature  and  at  210OC 
respectively,  at  a  level  of  8000  p.S,  located  by  the  peak  of  the  half-sine  clamp,  have 
been  carried  out.  Ultrasonic  scans  of  specimens  S3  and  S4  before  any  loading  cycles 
and  after  100,  500,  1000,  2000,  5000,  10000,  20000  and  50000  loading  cycles  are 
shown  respectively  in  Figures  l.a-h  and  Figures  2.a-h.  A  small  delamination, 
indicated  by  lighter  areas  in  the  scans,  starts  to  show  in  both  specimens  S3  and  S4 
after  applying  500  loading  cycles  and  increases  substantially  after  5000  loading 
cycles.  After  5000  loading  cycles,  the  damage  area  increased  more  for  specimen  S4 
than  specimen  S3,  which  shows  that  the  latter  is  slightly  more  fatigue  resistant.  In 
other  words,  when  increasing  the  temperature  from  250C  to  210<^C,  the  resistance  to 
fatigue  slightly  decreases.  Figures  3.a-h  and  Figures  4.a-h  show  the  ultrasonic  scans 
for  PMR15  specimens  PI  and  P2  before  and  after  several  loading  cycles.  For  both 
specimens  PI  and  P2,  damage  starts  after  500  loading  cycles  and  increases 
substantially  after  20000  loading  cycles.  At  this  stage,  delamination  areas  are  similar 
for  both  specimens  PI  and  P2  and  just  a  little  more  pronounced  in  specimen  P2, 
which  shows  that  the  latter  is  slightly  less  fatigue  resistant.  Hence,  an  increase  in 
temperature  leads  to  a  decrease  in  the  fatigue  resistance  properties  of  the  specimens. 
From  Figures  1,  2,  3  and  4,  one  can  conclude  that  the  PMR15  specimens  tested  were 
slightly  more  fatigue  resistant  than  the  SE300  specimens.  In  fact,  damage  in 
specimens  was  generally  more  defined,  clearer  and  spread  more  rapidly  in  the  SE300 
samples  than  was  the  case  for  the  PMR15  samples.  Figure  5  shows  damage 
propagation  occuring  in  an  XAS/914  sample  (XI),  with  (0o/±45®/0O)s  stacking 
sequence,  tested  at  8000  |iS  level  and  at  room  temperature,  from  [3].  Note  that 
substantial  damage  existed  after  1000  loading  cycles  in  this  specimen,  which  shows 
that  both  SE300  and  PMR15  are  more  fatigue  resistant  than  XAS/914  at  room 
temperature. 

Acoustic  fatigue  behaviour  of  panels  of  the  selected  materials 
Investigations  were  carried  out  by  installing  the  CFRP  plates  in  an  acoustic 
progressive  wave  tunnel,  (APWT)  in  order  to  determine  the  response  of  CFRP  plates 
under  broadband  acoustic  excitation  simulating  jet  noise.  The  plate  was  fully 
clamped  around  its  boundaries  on  to  a  vertical  steel  frame  fixed  to  one  side  of  the 
APWT,  so  that  it  formed  one  of  the  vertical  walls  of  the  test  section  of  the  APWT. 


983 


Overall  sound  pressure  levels  (OSPL)  up  to  165  dB  of  broadband  noise  in  the  test 
section  of  the  tunnel  was  generated  by  a  Wyle  Laboratories  WAS  3000  siren.  A 
heater  panel  capable  of  heating  and  maintaining  the  temperature  of  test  plates  up  to 

300*^0  while  mounted  on  the  tunnel  was  designed  and  built.  Temperatures  were 
monitored  and  controlled  via  thermocouples  on  the  panel.  Plates  were  excited  by 
broadband  excitation  in  the  frequency  range  80-800  Hz.  A  B&K  type  4136 
microphone  mounted  at  the  centre  of  the  test  section  of  the  tunnel  adjacent  to  the 
mid-point  of  the  plate  were  used  for  sound  pressure  measurements.  Eight  strain 
gauges,  four  on  each  side  of  the  plate  were  attached  in  order  to  monitor  the  strain 
distribution  in  the  panel  while  the  rig  was  running;  more  details  of  the  experiment 
may  be  found  in  [1].  Acoustic  tests  were  run,  at  various  temperatures  and  OSPL. 
Since  plates  were  excited  in  the  frequency  range  80-800  Hz,  spectral  analyses 
would  not  include  the  first  natural  frequency.  The  natural  fundamental  frequency  of 
an  SE300  clamped  panel  was  found  to  be  49.02  Hz  analytically.  The  second  -and 
third  resonance  frequencies  were  149.5  and  198.5  Hz,  as  calculated  from  strain 
spectral  densities,  from  tests  carried  out  with  an  OSPL  of  156  dB  and  a  temperature 
of  1620C.  At  162  dB,  results  showed  that  the  second  resonance  frequency  was  113 
Hz  at  150OC  and  106.5  Hz  at  195^0,  which  shows  that  when  the  temperature  and 
the  OSPL  increased,  the  resonance  frequencies  of  the  plates  decreased.  Also,  it  was 
found  that  the  damping  increased  at  elevated  temperatures.  The  overall  modal 
viscous  damping  ratios,  for  the  second  mode,  were  calculated  from  strain  spectral 
densities,  for  an  SE300  panel  driven  at  an  OSPL  of  162  dB  and  at  1950C,  and  was 
found  to  be  8.91%;  this  value  is  similar  to  that  calculated  from  analytical 
simulations,  for  the  first  mode,  which  is  8,50%  at  242^0,  (see  the  analytical  section 
below).  A  typical  strain  spectral  density  obtained  from  recorded  results  is  presented 
in  Figure  6  for  an  SE300  specimen,  at  an  OSPL  =  156  dB  and  at  1620C,  from  a 
strain  gauge  in  the  centre  of  the  specimen.  Maximum  RMS  strain  values  recorded 
from  experimental  tests,  at  a  strain  gauge  in  the  centre  of  the  specimen  were,  at  an 
OPSL  =156  dB,  1300  \iS,  1800  \iS  and  2800  pS  at  room  temperature  and  at  90^0 
and  1620C  respectively.  These  results  clearly  indicate  a  trend  for  significant 
increase  in  dynamic  response  with  increasing  temperature. 

Experiments  were  also  carried  out  on  a  PMR15  panel  at  various  OSPL  and 
temperatures.  Results  from  tests  run  at  159  dB  and  at  room  temperature  show  that 
the  second  and  third  resonance  frequencies  were  indicated  as  112  and  182.5  Hz. 
When  the  OSPL  increased  to  162  dB,  the  second  and  third  resonance  frequencies 
decreased  to  110.5  Hz  and  176.5  Hz.  At  the  same  OSPL  (162dB)  and  when 
temperature  increased  to  2810C,  the  third  resonance  frequency  became  139  Hz.  This 


984 


shows  that,  for  PMR15  plates,  when  the  temperature  and  the  OSPL  increased,  the 
resonance  frequencies  of  the  plates  decreased.  It  was  also  clear  that  modal  damping 
increased  with  increasing  temperature.  In  fact,  the  overall  viscous  damping  reached 
20%,  for  the  second  mode,  at  an  OSPL=162  dB  and  at  2810C.  It  must  be  stated  here, 
however,  that  apparent  damping  trends  could  include  nonlinear  effects  which 
influence  bandwidths  of  resonances.  Maximum  RMS  strain  values  recorded  at  room 
temperature,  by  a  strain  gauge  in  the  centre  of  the  plate  were  found  to  be:  2700  pS  at 
153  dB,  2800  pS  at  157.9  dB  and  2900  pS  at  159  dB. 

It  is  clear  from  these  values  that  increasing  the  OSPL  obviously  leads  to  an  increase 
of  the  strain  in  the  plate.  Similar  results  were  observed  when  the  temperature  was 
increased.  In  fact,  at  an  OSPL  of  162  dB,  the  maximum  strain  values  recorded  by  a 
strain  gauge  in  the  centre  of  the  specimen  were  3000  pS,  3400  pS  and  5000  pS  at 
105^0,  1650C  and  281^0  respectively,  which  clearly  indicates  the  effects  of 
temperature.  It  was  observed  that  both  the  PMR15  and  SE300  panels  behaved  in  a 
non  linear  manner. 

Attempts  to  acoustically  fatigue  a  PMR15  panel  were  made  at  162  dB.  No  signs  of 
fatigue  damage  were  shown  in  an  ultrasonic  scan  of  the  panel  after  1389  minutes  of 
running  time. 


3-Analytical  work 

In  order  to  examine  various  methods  for  including  damping  in  a  structure,  parametric 
studies  were  carried  out  using  the  finite  element  FE  method.  ANSYS  software  has 
been  used.  A  three-dimensional,  3D  layered  element,  SOLID46  was  used  to  build 
theoretical  models.  The  element  is  defined  by  eight  nodal  points,  average  layer 
thickness,  layer  material  direction  angles  and  orthotropic  material  properties,  [4]. 
Meshes  were  built  in  order  to  carry  out  modal  and  harmonic  analyses  of  multilayered 
composite  plates  (410  mm,  280  mm,  2  mm).  The  plates  were  fully  clamped  along  all 
edges,  in  order  to  simulate  the  panels  tested  in  the  APWT.  Natural  frequencies  were 
first  determined  from  free  vibration  analyses  and  compared  to  resonance  frequency 
values  derived  from  experimental  data.  Then,  the  plate  was  driven  by  harmonic 
loading  at  one  point  of  application.  The  forcing  frequency  varied  from  0  to  400  Hz. 
The  amplitude  of  the  load  was  50  N.  Results  for  displacements  and  response  phase 
angles  relative  to  the  force  for  a  chosen  position  on  the  plate  as  a  function  of 
frequency  were  obtained.  The  approach  was  then  to  carry  out  parameter  studies  in 
order  to  examine  various  methods  for  including  damping  in  the  structure,  i.e.  to  use 
highly  damped  matrix  material  throughout  the  whole  structure  or  possible 


985 


incorporation  in  a  few  layers.  Structural  damping  was  included,  allowing  models  to 
run  with  different  damping  values  in  each  ply  of  the  panel.  Structural  damping  is 
inherent  in  the  structure  and  depends  on  the  natural  frequency;  details  on  structural 
damping  modeling  may  be  found  in  [1].  Analyses  were  performed  considering 
structural  damping  for  the  first  mode.  The  structural  damping  was  then  varied  for 
plies  with  the  same  orientation  for  a  viscous  damping  ratio  ^  =  0.01, 0.02,  0.05,  0.10 
and  0.20. 

Simulations  with  SE300 

Models  were  built  up  with  the  following  stacking  sequence  ((45°/45°),(0°/90°))s, 
lay-up  used  for  the  experimental  plates.  Table  2  gives  the  first  three  modal 
frequencies  of  the  panel  obtained  from  free  vibration  analyses  results.  Harmonic 
simulations  were  carried  out  and  the  overall  damping  value  was  calculated  for  each 
case  with  results  given  in  Table  3.  As  can  be  seen,  if  high  overall  damping  is  needed 
for  a  structure  composed  of  the  SE300  material,  increasing  the  damping  value  of  the 
(45°/45®)  orientation  plies  most  significantly  increases  the  overall  damping  value  of 
the  panel.  In  fact,  putting  a  damping  value  of  20%  in  the  (45°/45°)  orientation  plies 
leads  to  an  overall  viscous  damping  value  of  14.52%,  which  is  better  than  including 
a  10%  damping  value  in  all  of  the  plies  of  the  structure. 

Harmonic  analyses  of  fully  clamped  plates  were  also  performed  with  the  values  of 
material  properties  taken  at  several  temperatures.  Simulations  were  carried  out  with 
material  properties  at  2420C  and  300OC.  Free  vibration  analyses  permitted 
calculation  of  the  modal  frequencies  of  the  panels  at  the  temperatures  mentioned 
above.  Table  2  also  lists  the  first  three  modal  frequencies  from  analyses  with 
material  properties  at  242^0  and  300^0.  The  overall  viscous  damping  values, 
obtained  from  FE  simulations,  are  given  for  each  ternperamre  in  Table  4.  Again,  the 
damping  value  has  been  varied  through  the  layers  and  the  overall  damping  value  was 
calculated  in  order  to  see  which  of  the  plies  contributes  the  most  to  heavily  damp  the 
plate.  It  was  found  that  putting  a  damping  value  of  20%  in  the  (45°/45°)  orientation 
plies,  the  first  mode  viscous  damping  ratios  were  14.62%  and  14.55%  at  242^0  and 
300OC  respectively.  This  shows  that  this  material  is  more  highly  damped  at  high 
temperature  and  presents  better  damping  properties  of  the  two  materials  at  242^0. 

Simulations  with  PMR15 

Free  vibration  analyses  of  models  built  up  with  the  following  stacking  sequence 
((45°/45°),(0°/90®))s  were  carried  out  and  the  first  three  modal  frequencies  of  the 
panel  are  shown  in  Table  5.  Harmonic  analyses  were  run  and  the  overall  damping 


986 


value  was  calculated  for  each  simulation  with  results  given  in  Table  6.  If  high 
overall  damping  is  needed  for  a  structure  composed  of  the  PMR15  material, 
increasing  the  damping  value  of  the  (45°/45®)  orientation  plies  most  significantly 
increases  the  overall  damping  value  of  the  panel.  In  fact,  putting  a  damping  value 
of  20%  in  the  (45°/45°)  orientation  plies  leads  to  an  overall  viscous  damping  value 
of  14.39%,  while  if  (0°/90®)  orientation  plies  have  a  20%  damping  value,  the 
overall  damping  is  7.42%. 

Harmonic  analyses  of  fully  clamped  plates  were  also  carried  out  with  the  values  of 
material  properties  taken  at  several  temperatures.  Simulations  were  carried  out 
using  material  properties  at  3750C  and  400oC.  Free  vibration  analyses  permitted 
calculation  of  the  modal  frequencies  of  the  panels  at  the  temperatures  mentioned 
above.  Table  5  lists  the  first  three  modal  frequencies  from  analyses  with  material 
properties  at  3750C  and  400^0.  The  overall  viscous  damping  values,  obtained 
from  FE  simulations,  are  given  for  each  temperature  in  Table  7.  Again,  the 
damping  value  has  been  varied  through  the  layers  and  the  overall  damping  value 
was  calculated  in  order  to  see  which  of  the  plies  contributes  the  most  to  heavily 

damp  the  plate.  It  was  found  that  putting  a  damping  value  of  20%  in  the  (45‘^/45°) 
orientation  plies,  the  first  mode  viscous  damping  ratios  were  18.39%  and  16.94% 
at  3750c  and  400OC  respectively.  This  shows  that  this  material  is  more  highly 
damped  at  high  temperature  and  presents  better  damping  properties  of  the  two 

materials  at  3750C. 


4~Conclusions 

Two  matrix  materials,  SE300  and  PMR15,  with  potential  for  use  in  aircraft 

structures  in  a  severe  environment,  i.e.  temperatures  up  to  SOO^C  were  selected  for 
this  study.  Material  properties  were  determined  using  DMTA  techniques  and 
results  show  that  these  materials  have  high  damping  abilities  at  high  temperature. 
Dynamic  loading  tests,  performed  in  flexure  at  room  and  high  temperature  showed 
that  the  carbon  fibre  reinforced  PMR15  material  is  more  fatigue  resistant  than 
SE300  and  XAS/914  based  composites.  Acoustic  tests  using  a  progressive  wave 
tunnel,  up  to  a  random  acoustic  OSPL  of  162  dB,  at  room  temperature  and 
elevated  temperamres  up  to  2810C  were  also  performed.  When  increasing  the 
excitation  level  and  the  temperature  higher  strain  values  in  the  centre  of  the  panels 
were  recorded.  Free  vibration  and  harmonic  FE  analyses  permitted  determination 
of  the  natural  frequencies  and  the  overall  viscous  damping  values.  Resonance 
frequencies  determined  from  results  obtained  from  acoustic  tests  were  similar  to 


987 


natural  frequencies  obtained  from  FE  simulations.  Overall  viscous  damping  values 
obtained  from  experimental  results  agreed  well  with  those  obtained  from  the  FE 
analyses  for  SE300  panels.  Results  obtained  for  PMR15  panels,  from  tests,  were 
higher  than  those  calculated  analytically.  Both  tests  and  simulations  showed  that 
SE300  and  PMR15  present  higher  damping  capabilies  at  high  temperatures. 
Conclusions,  via  parameter  studies  including  material  damping,  have  been  drawn 
concerning  optimal  incorporation  of  a  highly  damped  matrix  material  into  a  high 
performance  structure. 


5-AcknowIegments 

The  authors  wish  to  thank  the  Minister  of  Defence  for  sponsorship  of  the  programme 
of  research  under  which  the  work  was  carried  out.  Thanks  are  also  due  to  Dr  M.  Nash 
of  the  DRA,  Famborough  for  many  helpful  discussions  throughout  the  project.  _ 

6-References 

1-  Benchekchou,  B.  and  White,  R.G.,  Acoustic  fatigue  and  damping  technology  in 
FRP  composites,  submitted  to  Composite  Structures. 

2-  Benchekchou,  B.  and  White,  R.G.,  Stresses  around  fasteners  in  composite 
structures  in  flexure  and  effects  on  fatigue  damage  initiation:  I-Cheese-head  bolts. 
Composite  structures,  33(2),  pp.  95-108,  November  1995. 

3-  Drew,  R.C.  and  White,  R.G.,  An  experimental  investigation  into  damage 
propagation  and  its  effects  upon  dynamic  properties  in  CFRP  composite  material  . 
Proceedings  of  the  Fourth  International  Conference  on  Composite  Structures,  Paisley 
College  of  Technology,  July  1987. 

4-  ANSYS  theoretical  manual,  Swanson  Analysis  Systems  Inc,  December  1992. 


988 


Table  La:  Loss  factor  and  Young’s  modulus  values  at  40^C  and  at  Tg  for 
SE300  samples  analysed  by  the  DMTA. 


stacking  sequences 

(4.45/-45)4 

(-457+45)4 

(45/45;0/90)s 

(0/90)4 

Tg(°C) 

242 

242 

240.3 

238.71 

T|  atTg 

0.085 

0.085 

0.081 

0.061 

Ti  at  40°C 

0.012 

0.014 

0.010 

0.0097 

Log  E’  at40°C 

9.870 

9.840 

9.970 

10:097 

Table  Lb:  Loss  factor  and  Young’s  modulus  values  at  40^C  and  at  Tg  for 
PMR15  samples  analysed  by  the  DMTA. 


Stacking  sequences 

(+45/--45)4 

(0/90;45/45)s 

(45/45;0/90)s 

(0/90)4 

Tg  (OC) 

372 

375 

375 

375 

Tl  atTg 

0.117 

0.124 

0.121 

0.129 

T|  at  40®C 

0.0110 

0.0138 

0.0086 

0.0117 

Log  E’  at  40OC 

9.583 

9.944 

9.972 

9.875 

989 


Table  2:  The  first  three  modal  frequencies  for  SE300  panel;  analyses  carried  out 
with  material  propert  ies  at  room  temperature,  at  242*^0  and  at  300°C. 


Room  temperature 

242^0 

300OC 

49.02Hz 

44.72Hz 

43.61Hz 

155.80Hz 

143.27Hz 

140.17Hz 

212.04Hz 

194.19Hz 

189.68Hz 

Table  3:  Overall  viscous  damping  values  of  SE300  panel.  Values  are  calculated  from 
results  obtained  from  harmonic  analyses;  the  material  damping  being  considered  for 
the  first  mode. 


Simulation  with 
damping  of 

(45°/45°)orientation 

plies 

(0®/90°)orientation 

plies 

5% 

3.81% 

2.51% 

10% 

7.06% 

3.92% 

20% 

14.52% 

11.57% 

Table  4:  Overall  viscous  damping  values  of  SE300  panel,  for  the  first  mode. 


Temperature 

(°C) 

Overall  viscous  damping 

25 

1.20% 

242 

8.50% 

300 

5.45% 

990 


Table  5:  The  first  three  modal  frequencies  for  PMR15  panel;  FE  analyses  carried  out 
with  material  properties  at  room  temperature,  375°C  and  400°C. 


Room  temperature 

3750c 

400OC 

43.43Hz 

32.68Hz 

25.54Hz 

130.35Hz 

99.65Hz 

78.83Hz 

183.96Hz 

139.04Hz 

109.11Hz 

Table  6;  Overall  viscous  damping  values  of  PMR15  panel.  Values  are  calculated 
from  results  obtained  from  harmonic  analyses;  the  material  damping  being 
considered  for  the  first  mode.. 


Simulation  with 
damping  of 

(45°/450)orientation 

plies 

(0o/90O)orientation 

plies 

5% 

3.76% 

2.44% 

10% 

7.26% 

4.10% 

20% 

14.39% 

7.42% 

Table  7:  Overall  viscous  damping  values  of  PMR15  panel,  for  the  first  mode.. 


Temperature 

Overall  viscous  damping 

(OQ 

25 

1.33% 

275 

13.24% 

400 

8.6% 

991 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  1000  loading  cycles  e:  after  2000  loading  cycles  f:  after  5000  loading  cycles 


g:  after  1 0000  loading  cycles  h:  after  20000  loading  cycles 


Figure  1.  Ultrasonic  scans  of  specimen  S3  after  applying  different  numbers  of  loading  cycles. 
(SE  300  material,  ambient  temperature) 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  1000  loading  cycles  e:  after  2000  loading  cycles  f:  after  5000  loading  cycles 


Fisure  2.  Ultrasonic  scans  of  specimen  S4  after  applying  different  numbers  of  loading  cycles. 

(SE300  material,  210OC) 


992 


a:  before  any  loading  cycles  b:  after  100  loading  cycles  c:  after  500  loading  cycles 


d:  after  2000  loading  cycles  e:  after  20000  loading  cycles  f:  after  50000  loading  cycles 


g:  after  100000  loading  cycles 


Figure  3.  Ultrasonic  scans  of  specimen  PI  after  applying  different  numbers  of  loading  cycles 
(PMR15  material,  ambient  temperature,  8000]LlS) 


a:  before  any  loading  cycles  b:  after  500  loading  cycles  c:  after  1000  loading  cycles 


g:  after  10000  loading  cycles 


h:  after  20000  loading  cycles 


Figure  4.  Ultrasonic  scans  of  specimen  P2  after  applying  different  numbers  of  loading  cycles. 

(PMR15  material,  2100C,  8000M.S) 


993 


a:  before  any  b:  after  100  c:  after  500  d:  after  1000 

loading  cycles  loading  cycles  loading  cycles  loading  cycles 


e:  after  2000  f:  after  5000  g:  after  10000  h:  after  20000 

loading  cycles  loading  cycles  loading  cycles  loading  cycles 

Figure  5.  Ultrasonic  scans  of  an  X AS/9 14  specimen  fatigued  at  a  level  of  8000  llS 
showing  the  damage  propagation;  the  lay-up  is  (0/±45/90)s,  [3], 


0  Lin  Hz  RCLD  1.6k 


Figure  6:  SE300  specimen  SI  strain  spectral  density,  recorded  from  strain  gauge 
ST2,  OSPL  =156  dB,  temperature  =  162^C. 


994 


THE  BEHAVIOUR  OF  LIGHT  WEIGHT  HONEYCOMB  SANDWICH 
PANELS  UNDER  ACOUSTIC  LOADING 


David  Millar 
Senior  Stress  Engineer 
Short  Bros.  PLC 
Airport  Road 
Belfast 

Northern  Ireland 


SUMMARY 

This  paper  discusses  the  results  of  a  progressive  wave  tube  test  on  a  carbon  composite 
honeycomb  sandwich  panel.  A  comparison  was  made  with  the  test  panel  failure  and  the 
failure  of  panels  of  similar  construction  used  in  the  intake  ducts  of  jet  engine  nacelles. 
The  measured  panel  response  is  compared  with  traditional  analytical  methods  and  finite 
element  techniques. 


Nomenclature 

=  Overall  rms  stress  (psi)  or  strain  (jxs). 

7t  =3.14159 

=  Fundamental  frequency  (Hz). 

5  =  Critical  damping  ratio  (*0.017). 

Lps(fn)  =  Spectrum  level  of  acoustic  pressure  (-  expressed  as  a  fluctuating  rms  pressure  in  psi  in  a 
1  Hz  band). 

jr  =  Joint  acceptance  function  (non  dimensional). 

=  Characteristic  modal  pressure  (psi) 
ph  =  Mass  per  unit  area  (Ib/in^) 

Sic  “  Modal  stress  (psi)  or  modal  strain  (|j£). 

Wjc  =  Modal  displacement  (in), 

a  =  Panel  length  (in), 

b  =  Panel  width  (in). 

x,y,z  =  Co-ordinate  axes. 


1.0  Introduction 

Honeycomb  sandwich  panels  have  been  used  for  some  time  in  the  aircraft  industry  as 
structural  members  which  offer  a  high  bending  stiffness  relative  to  their  weight.  In 
particular,  they  have  proved  very  attractive  in  the  construction  of  jet  engine  nacelle 
intake  ducts  where,  in  addition  to  their  load  carrying  ability,  they  have  been  used  for 
noise  attenuation. 


995 


2.0  Acoustic  Fatigue 

The  intake  duct  of  a  jet  engine  nacelle  can  experience  a  severe  acoustic  environment 
and  as  such  the  integrity  of  the  nacelle  must  be  assessed  with  regard  to  acoustic  fatigue 
[1],  Acoustic  fatigue  characterises  the  behaviour  of  structures  subject  to  acoustic 
loading,  in  which  the  fluctuating  sound  pressure  levels  can  lead  to  a  fatigue  failure  of 
the  structure.  The  traditional  approach  to  acoustic  fatigue  analysis  has  assumed 
fundamental  mode  response  and  given  that  aircraft  panels  will  in  general,  have 
fundamental  frequencies  of  the  order  of  several  hundred  hertz,  it  is  clear  that  the 
potential  to  accumulate  several  thousand  fatigue  cycles  per  flight  can  exist. 

Techniques  for  analysing  the  response  of  structures  to  acoustic  loads  were  developed 
originally  by  Miles  [2]  and  Powell  [3],  Other  significant  contributions  are  listed  in 
References  4-7.  Design  guides  such  as  AGARD  [8]  and  the  Engineering  Sciences 
Data  Unit  (ESDU)  series  of  data  sheets  on  vibration  and  acoustic  fatigue  [9],  have 
proved  useful  in  the  early  stages  of  design. 

Note  -  further  details  on  the  general  subject  of  acoustic  fatigue  can  be  found  in  Ref 
10,  while  a  more  detailed  review  of  the  subject  up  to  more  recent  times  is  presented  in 
Ref  11. 


3.0  In  Service  Failures 

In  recent  years  a  number  of  failures  have  been  experienced  involving  intake  barrel 
honeycomb  sandwich  panels.  Failures  have  been  experienced  with  panels  which  had 
both  aluminium  facing  and  backing  skins  and  carbon  composite  panels.  The  metal 
intake  liner  was  observed  to  have  skin  cracking  and  also  core  failure,  while  the 
composite  panel  was  only  observed  to  have  core  failure. 

With  regard  to  the  metal  panels,  flight  testing  was  carried  out  and  the  predominant 
response  frequency  was  observed  to  be  at  the  fan  blade  passing  frequency  -  much 
higher  than  the  fundamental  frequency  of  the  intake  barrel;  this  went  some  way  to 
explaining  why  the  traditional  approach  in  estimating  the  response  did  not  indicate  a 
cause  for  concern.  The  response  of  the  panel  was  also  very  narrow  band  -  almost  a 
pure  sinusoid  (again  differing  from  the  traditional  approach  of  broad  band/random  load 
and  response),  and  the  subsequent  analysis  of  the  results  was  based  on  a  mechanical 
fatigue  approach  [12].  Subsequent  fleet  inspections  revealed  that  core  failure  was 
observed  prior  to  skin  failure  and  it  was  assumed  that  the  skin  failure  was  in  fact 
caused  by  a  breakdown  in  the  sandwich  panel  construction.  The  core  was  replaced 
with  a  higher  density  variety,  with  higher  shear  strength  and  moduli.  This  modification 
has  been  in  service  for  several  years  with  no  reported  failures.  The  modification 
represented  only  a  moderate  weight  increase  of  the  panel,  without  recourse  to 
changing  skin  thickness,  which  would  have  proved  very  expensive  and  resulted  in  a 
substantial  weight  penalty. 

As  mentioned  above,  another  intake  duct,  of  carbon  composite  construction,  also 
began  to  suffer  from  core  failure.  The  panels  of  this  duct  had  a  carbon  backing  skin 
while  the  facing  skin  had  a  wire  mesh  bonded  to  an  open  weave  carbon  sheet.  The  only 
similarity  was  the  use  of  the  same  density  of  honeycomb  core  (although  of  different 
cell  size  and  depth).  For  other  reasons  this  core  had  been  replaced  by  a  heavier  variety, 
prior  to  the  discovery  of  the  core  failures  and  the  impact  of  the  failures  was  minimised. 


996 


Limited  data  is  available  on  similar  failures  and  only  2  other  cases,  regarding  nacelle 
intake  barrels,  appear  to  have  been  documented  [13  &  14],  however  neither  case 
involved  sandwich  panels. 

A  number  of  theories  had  been  put  forward  as  to  the  cause  of  the  failures.  These 
included  neighbouring  cells  resonating  out  of  phase,  cell  walls  resonating  or  possibly 
the  panel  vibrating  as  a  2  degree  of  freedom  system  (the  facing  and  backing  skins 
acting  as  the  masses,  with  the  core  as  the  spring)  -  this  phenomenon  had  originally 
been  investigated  by  Mead  [15]. 


4.0  Physical  Testing 

A  number  of  tests  were  carried  out  with  "beam"  type  high  cycle  fatigue  specimens  and 
also  small  segments  of  intake  barrel.  None  of  these  tests  were  able  to  reproduce  the 
failures  observed  in  service  (Figure  1.0)  which  further  served  to  reinforce  the  belief 
that  the  failures  were  attributed  to  an  acoustic  mechanism  as  opposed  to  a  mechanical 
vibration  mechanism,  however  in  an  attempt  to  cover  all  aspects  it  was  decided  to 
carry  out  a  progressive  wave  tube  (PWT)  test  on  an  abbreviated  panel. 

For  simplicity  it  was  decided  to  test  a  flat  sandwich  panel  of  overall  dimensions 
36"x21"  (Figure  2.0).  The  panel  was  instrumented  with  12  strain  gauges  and  2 
accelerometers.  Two  pressure  transducers  were  also  mounted  in  the  fixture 
surrounding  the  specimen. 

Testing  was  carried  out  by  the  Consultancy  Service  at  the  Institute  of  Sound  and 
Vibration  Research  (IS  VR)  at  the  University  of  Southampton. 


4.1  PWT  Results 

The  panel  was  first  subject  to  a  sine  sweep  from  50  to  1000  Hz  in  order  to  identify  its 
resonant  frequencies.  The  response  of  a  strain  gauge  at  the  centre  of  the  panel  has  been 
included  in  Figure  3.0.  On  completion  of  the  sine  sweeps,  the  linearity  tests  were 
carried  out..  As  only  8  channels  could  be  accommodated  at  one  time,  it  had  been 
decided  to  arrange  the  parameters  into  5  groups,  with  each  group  containing  4  strain 
gauges,  2  accelerometers,  1  pressure  transducer  on  the  fixture  and  1  pressure 
transducer  in  the  PWT  (this  was  required  by  the  facility  for  the  feedback  loop). 


The  initial  tests  were  carried  out  with  a  power  spectral  density  of  the  applied  loading 
constant  over  the  100  Hz  to  500  Hz  range,  however  when  using  this  bandwidth  only 
155  dB  overall,  could  be  achieved.  In  an  attempt  to  increase  the  strain  levels  it  was 
decided  to  reduce  the  bandwidth  to  200  Hz.  The  bandwidth  (BW)  was  subsequently 
reduced  to  100  Hz  and  finally  1/3  octave  centred  on  the  predominant  response 
frequency  of  the  panel.  When  failure  occurred  a  dramatic  change  in  response  was 
observed.  The  failure  mechanism  was  that  of  core  failure  as  shown  in  Figure  4.0.  There 
was  no  indication  of  facing  or  backing  skin  distress. 


997 


5.0  Comparison  With  Theoretical  Predictions 
5.1  Fundamental  Frequency 

From  the  strain  gauge  readings  the  panel  was  seen  to  be  vibrating  with  simply 
supported  edge  conditions.  Soovere  [7]  suggests  that  "effective"  dimensions 
(essentially  from  the  start  of  the  pan  down)  be  used  to  determine  the  fundamental 
frequency  which  is  given  by; 


x\n 


This  equation  is  applicable  to  simply  supported  panels  with  isotropic  facing  and 
backing  skins,  thus  for  the  purpose  of  applying  the  above  equation,  the  actual  section 
was  approximated  to  a  symmetric  (isotropic)  section.  The  predicted  fundamental 
frequency  is  given  below.  It  was  observed  however,  that  if  the  panel  dimensions  are 
taken  relative  to  mid  way  between  the  staggered  pitch  of  the  fasteners  a  significant 
improvement  was  achieved  (see  "Soovere  (2)"  in  table  1).  Alternative  frequency 
estimations  using  an  FE  model  and  an  ESDU  data  item  [16]  are  summarised  in  the 
following  table; 


Method 

Freq.(Hz) 

%  Error 

Measured 

228 

- 

Soovere 

274.3 

+20.3 

Soovere 

(2) 

213.3 

-6.4 

FE 

239.04 

+4.8 

ESDU 

193 

-15.3 

Table  1  -  Comparison  of  calculated  frequencies  for  simply  supported  sandwich 

panel. 

Note;  the  percentage  error  is  based  on  the  actual  measured  response  frequency  of  the 
panel  in  the  PWT. 

Given  that  the  excitation  bandwidth  extended  (at  least  initially)  up  to  500  Hz,  modes 
up  to  500  Hz  were  obtained  from  the  FE  model.  In  actual  fact  2  FE  models  were  used, 
the  first  was  a  basic  model  with  380  elements,  however  a  more  detailed  model,  shown 
in  Figure  5.0,  (with  essentially  each  element  split  into  4)  was  used  for  the  results 
presented  in  this  paper.  The  predicted  modes  from  the  FE  model  were  as  follows; 


Mode 

No. 

Frequency 

(Hz) 

Mode  No.  in  x 
direction  (m) 

Mode  No.  in  y 
direction  (n) 

Figure 

No. 

1 

239.04 

1 

1 

6 

2 

334.0 

2 

1 

7 

3 

430.02 

1 

2 

8 

4 

1 

9 

Table  2  -  Finite  Element  Model  Predicted  Frequencies. 


998 


5.2  rms  Strain 

The  predicted  strains  were  calculated  using  Blevins'  normal  mode  method  (NMM)  [5], 
with  a  joint  acceptance  of  unity  for  the  fundamental  mode  of  vibration,  using  the 
following  expression; 


s^=  Lp.(fJ  .  ^  (2) 

In  an  attempt  to  improve  the  estimated  response,  the  rms  strain  was  calculated  for 
each  mode  within  the  bandwidth  of  excitation.  The  Joint  accetptance  for  each  mode 
was  calculated  using  equation  3  and  the  calculated  strains  for  each  mode  were  then 
factored  by  the  relevant  joint  acceltance  term.  The  overall  strain  was  then  calculated 
for  all  the  relevant  modes.  A  comparison  with  ESDI!  [16]  has  also  been  included, 
however  the  ESDU  method  does  not  provide  an  indication  of  shear  stress  in  the  core. 

Soovere  presents  a  simple  expression  for  the  joint  acceptance  function  for  a  simply 
supported  panel  excited  by  an  (acoustic)  progressive  wave,  for  the  case  where  n  is 
odd; 

•2  _  ^  (l-Cos(m7c)Cos(c0ra/c)  ,  . 

(1  -  (cD^a/ mrcc)^) 

Note,  when  n  is  even  the  joint  acceptance  is  zero. 

Given  that  the  bandwidth  varied  for  the  applied  loading,  the  overall  SPLs  were 
expressed  as  spectrum  levels  for  the  purpose  of  comparison  in  the  linearity  results,  the 
results  (both  measured  and  predicted)  have  been  summarised  in  table  3,  (SGI  results 
have  been  plotted  in  Figure  10.0).  The  results  from  the  ESDU  data  item  [16]  have 
been  included  in  table  4  for  comparison. 

Note  -  due  to  recorder  channel  limitations  SGI  &  SG2  were  not  connected  at  the  time 
of  failure  and  no  results  were  available  at  the  highest  sound  pressure  levels. 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Measured 
Strains(u£) 
SGI  SG2 

Calculated 
(ps)  j=l 

SGI  SG2 

Calculated 
(Multi  Mode) 
SGI  (us)  SG2 

130 

107 

7 

7 

8.7 

9.2 

2.5 

2.6 

140 

117 

20 

19 

27.5 

29.1 

8.0 

8.1 

150 

127 

55 

60 

87.0 

92.0 

25.4 

25.8 

155 

132 

100 

100 

154.7 

163.7 

45.1 

45.8 

157 

134 

130 

130 

194.8 

206.0 

56.8 

57.6 

163 

140 

- 

- 

388.7 

411.1 

202.5 

162.7 

164 

141 

- 

- 

436.1 

461.2 

312.0 

250.7 

Table  3  -  Comparison  of  Measured  &  Predicted  rms  Strains 
for  the  Panel  Centre,  Facing  &  Backing  Skin  Gauges. 


999 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Measured 
Strains  (pe) 
SGI  SG2 

ESDI!  Strains 

SGI  SG2 

130 

107 

7 

7 

10.1 

29 

140 

117 

20 

19 

150 

127 

55 

60 

101 

290 

155 

132 

100 

179.5 

515.7 

157 

134 

130 

■E9 

253.6 

728.5 

163 

140 

- 

- 

637 

1830 

164 

141 

- 

- 

1010 

2900 

Table  4  -  Comparison  of  Measured  &  ESDI!  Predictions  of  the 
rms  Strains  for  the  Panel  Centre,  Facing  &  Backing  Skin  Gauges. 

There  is  a  considerable  difference  in  the  calculated  response  from  using  a  joint 
acceptance  of  unity  for  the  fundamental  mode  and  that  when  estimating  the  joint 
acceptance  for  each  mode  and  calculating  the  overall  response  for  several  modes, 
however  it  was  observed  that  if  the  average  value  from  both  methods  is  used  the 
response  compares  favourably  with  that  measured  (-at  least  for  the  cases  under 
consideration).  The  average  value  has  been  included  on  the  linearity  plot  for  SGI, 
shown  in  Figure  10.0). In  general,  the  level  of  agreement  between  theory  and  practice 
was  considered  adequate  and  it  was  decided  to  apply  the  theory  to  estimating  the  shear 
stresses  in  the  core  (Table  5); 


OASPL 

(dB) 

Spectrum 
Level  SPL 
(dB) 

Core  Shear 
Stress  (J=l) 
(rms  psi) 

Core  Shear 
Stress 

(Multi  Mode) 
(rms  psi) 

Average 
Core  Shear 
Stress 
(rms  psi) 

Peak  Core 
Shear 
Stress 

_ (E£!i _ 

130 

107 

0.3 

0.09 

0.19 

0.58 

140 

117 

0.94 

0.28 

1.84 

150 

127 

2.99 

0.88 

1.93 

5.8 

155 

5.31 

1.57 

3.44 

10.32 

157 

134 

6.69 

1.97 

4.33 

13.0 

163 

140 

13.35 

5.57 

9.46 

28.38 

164 

141 

14.98 

8.59 

11.78 

35.34 

Table  5  -  Predicted  Core  Shear  Stress. 

6.0  Discussion  &  Recommendations 

The  ESDI!  method  proved  very  conservative  and  will  thus  give  a  degree  of 
confidence  when  used  in  the  early  stages  of  the  design  process.  Blevins  Normal  Mode 
Method  was  observed  to  give  reasonable  accuracy  in  predicting  the  highest  strains  in 
the  panel  and  would  merit  use  when  designs  have  been  fixed  to  some  degree;  at  which 
stage  FE  models  become  available. 

For  panels  whose  predominant  response  is  in  the  fundamental  mode  it  is  accepted  that 
the  contribution  from  shear  to  overall  deformation  is  very  small.  The  main  concern 
when  designing  a  honeycomb  sandwich  panel  which  is  subject  to  "severe"  acoustic 


1000 


loads  has  tended  to  focus  on  skin  strains  and  to  some  degree  the  properties  of  the  core 
material  have  been  ignored.  The  fact  that  low  skin  strains  are  observed  has  the  effect  of 
giving  an  impression  that  there  is  no  cause  for  concern,  however  when  the  properties 
of  the  core  material  are  low  or  unknown,  some  caution  is  required.  There  is 
unfortunately  no  available  S-N  data  for  the  type  of  honeycomb  used  in  the  construction 
of  the  panel,  however  the  allowable  ultimate  strength  for  the  core  material  is  of  the 
order  of  26  psi,  so  clearly  the  163  dB  level  was  sufficient  to  cause  a  static  failure  while 
the  lower  SPLs  can  be  assumed  to  the  have  contributed  to  initiating  fatigue  damage. 

On  cutting  up  the  test  panel,  a  large  disbond  was  observed  however  it  did  not  extend 
to  the  panel  edge  where  cracking  had  occurred  (the  mid  point  of  the  long  edge  being 
the  location  of  maximum  shear  for  a  simply  supported  panel)  and  it  was  the  opinion  of 
the  materials  department  that  the  failure  had  not  initiated  in  the  disbond. 

The  SPLs  used  in  the  test  were  not  excessively  high  and  were  comparable  to  service 
environments  (an  example  of  which  is  given  in  Table  6).  It  should  be  noted  that  while 
the  levels  in  Table  5  are  1/3  octave  bandwidths,  the  actual  spectrum  is  not  generally 
flat  within  each  band  for  engine  intakes,  but  is  rather  made  up  of  tones  (Figure  11). 
These  tones  or  spectrum  levels  can  thus  essentially  be  the  band  level  and  thus  some 
caution  should  be  exercised  when  converting  intake  band  levels  to  spectrum  levels 
using  the  traditional  approach  [17]. 


1/3  Octave  Centre 

Sound  Pressure 

Frequency  (Hz) 

Level  (dB) 

100 

141 

125 

133 

160 

140 

200 

142 

250 

140 

315 

139 

Table  6  -  Typical  Acoustic  Service  Environment. 

Note;  Overall  levels  may  reach  160  -  170  dB,  however  they  tend  to  be  influenced  by 
SPLs  at  blade  passing  frequencies,  which  are  much  higher  than  panel  fundamental 
frequencies. 


7.0  Conclusion 

It  has  been  shown  that  although  moderate  levels  of  acoustic  excitation  produce  quite 
low  overall  rms  strains  in  the  skins  of  honeycomb  sandwich  panels,  it  is  still  possible, 
when  using  very  light  weight  cores,  to  generate  core  shear  stresses  of  a  similar  order  of 
magnitude  to  the  allowable  ultimate  strength  of  the  material. 


Acknowledgements 

The  author  acknowledges  the  support  of  Short  Bros.  PLC  in  the  course  of  preparing 
this  paper  and  also  the  assistance  of  Mr  Neil  McWilliam  with  regard  to  the  FE 
modelling. 


1001 


References 


I .  0  Air  worthiness  Requirements  (JAR/FAR)  Section  25 .57 1  .d. 

2.0  Miles,  J.W.,  "On  Stmctural  Fatigue  Under  Random  Loading, "  Journal  of  the 
Aeronautical  Sciences,  (1954),Vol.21,  p753  -  762. 

3.0  Powell,  A.,  "On  the  Fatigue  Failure  of  Structures  due  to  Vibrations  Excited  by 
Random  Pressure  Fields,”  Journal  of  the  Acoustical  Society  of  America,  (1958), 
Vol.30,  No.l2,pll30-  1135. 

4.0  Clarkson,  B.L.,  "Stresses  in  Skin  Panels  Subjected  to  Random  Acoustic 
Loading,"  Journal  of  the  Royal  Aeronautical  Society,  (1968),  Vol.72, 
plOOO-  1010. 

5.0  Blevins,  R.D.,  "An  Approximate  Method  for  Sonic  Fatigue  Analysis  of 
Plates  &  Shells,"  Journal  of  Sound  &  Vibration,  (1989),  Vol.129,  No.l, 
p51-71. 

6.0  Holehouse,  I.,  "Sonic  Fatigue  Design  Techniques  for  Advanced  Composite 
Aircraft  Structures,"  AFWAL  TR  80-3019,(1980). 

7.0  Soovere,  J.,  "Random  Vibration  Analysis  of  Stiffened  Honeycomb  Panels  with 
Beveled  Edges,"  Journal  of  Aircraft,  (1986),  Vol.23,  No. 6,  p537-544. 

8.0  Acoustic  Fatigue  Design  Data  (Part  1),  AGARD-AG- 162-72,  (1972). 

9.0  ESDU  International,  London,  Series  on  Vibration  &  Acoustic  Fatigue. 

10.0  Richards,  E.J.,  Mead,  D.J., "Noise  and  Acoustic  Fatigue  in  Aeronautics,"  John 
Wiley  &  Sons,  New  York,  (1968). 

II. 0  Clarkson,  B.L.,  "A  Review  of  Sonic  Fatigue  Technology,"  NASA  CR  4587, 

(1995). 

12.0  Millar,  D.,  "Analysis  of  a  Honeycomb  Sandwich  Panel  Failure,"  M.Sc.  Thesis, 
University  of  Sheffield,  (1995). 

13.0  Holehouse,  L,  "Sonic  Fatigue  of  Aircraft  Structures  due  to  Jet  Engine  Fan 
Noise,"  Journal  of  Sound  &  Vibration,  (1971),  Vol.  17,  No. 3,  p287-298. 

14.0  Soovere,  J.,  "Correlation  of  Sonic  Fatigue  Failures  in  Large  Fan  Engine  Ducts 
with  Simplified  Theory,"  AGAEUD  CPI  13  (Symposium  on  Acoustic  Fatigue), 
(1972),  pi  1-1  -  11-13. 

15.0  Mead,  D.J.,  "Bond  Stresses  in  a  Randomly  Vibrating  Sandwich  Plate:  Single 
Mode  Theoiy,"  Journal  of  Sound  &  Vibration,  (1964),  Vol.l,  No. 3, 
p258-269. 

16.0  ESDU  Data  Item  86024  (ESDUpac  A8624),  "Estimation  of  RMS  Strain  in 
Laminated  Face  Plates  of  Simply  Supported  Sandwich  Panels  Subjected  to 
Random  Acoustic  Loading,"  Vol.  3  of  Vibration  &  Acoustic  Fatigue  Series. 

17.0  ESDU  Data  Item  66016,  "Bandwidth  Correction,"  Vol.  1  of  Vibration  & 
Acoustic  Fatigue  Series. 


1002 


Backing  Skin 


Figure  2.0  -  PWT  Test  Specimen. 


1003 


Microstrain  (dB) 


Figure  3.0  -  Response  of  Strain  Gauge  SGI  During  Sine  Sweep. 


Figure  4.0  -  Section  Through  Failure  Region  in  PWT  Panel. 


1004 


Figure  5.0  -  PWT  Panel  Finite  Element  Model. 


Figure  8.0  -  FE  Mode  1  (m==l,  n=2)  Figure  9.0  -  FE  Mode  2  (m=3,  n=l) 


1005 


Sound  Pressure  Level  (dB) 


Figure  10.0  -  Linearity  Plot  for  SGI  (Measured  &  NMM  Prediction). 


Figure  11.0  -  Typical  Spectral  Content  of  Intake  Duct  Sound 
Pressures  with  Equivalent  1/3  Octave  Levels  Superimposed. 


1006 


Time  Domain  Dynamic  Finite  Element  Modelling  in  Acoustic  Fatigue  Design 


Authors: 

P.  D.  Green 
Military  Aircraft 
British  Aerospace 
Warton 

A.  Killey 

Sowerby  Research  Centre 
British  Aerospace 
Filton 

Summary 

Advanced  Aircraft  are  expected  to  fly  in  increasingly  severe  and  varied  acoustic  environments. 
Improvements  are  needed  in  the  methods  used  to  design  aircraft  against  acoustic  fatigue.  Since  fatigue 
life  depends  strongly  on  the  magnitude  of  the  cyclic  stress  and  the  mean  stress,  it  is  important  to  be  able 
to  the  predict  the  dynamic  stress  response  of  an  aircraft  to  random  acoustic  loading  as  accurately  as 
possible. 

The  established  method  of  determining  fatigue  life  relies  on  linear  vibration  theory  and  assumes  that  the 
acoustic  pressure  is  fully  spatially  correlated  across  the  whole  structure.  The  technique  becomes 
increasingly  unsatisfactory  when  geometric  non-linearities  start  to  occur  at  lugh  noise  levels  and/or 
when  the  structure  is  significantly  curved.  Also  the  excitation  is  generally  not  in  phase  across  the  whole 
structure  because  of  complex  aerodynamic  effects. 

Recent  advances  in  finite  element  modelling,  combined  with  the  general  availability  of  extremely  fast 
supercomputers,  have  made  it  practical  to  carry  out  non-linear  random  vibration  response  predictions 
using  time  stepping  finite  element  (FE)  codes. 

Using  the  time  domain  Monte  Carlo  (TDMC)  technique  it  is  possible  to  model  multi-modal  vibrations 
of  stiffened  aircraft  panels  without  making  the  simplifying  assumptions  concerning  the  linearity  of  the 
response  and  the  characteristics  of  the  noise  excitation. 

The  technique  has  been  developed  initially  using  a  simple  flat  plate  model.  This  paper  presents  some  of 
the  results  obtained  during  the  course  of  this  work.  Also  described  are  the  results  of  a  study  of  the 
“snap-through”  behaviour  of  the  flat  plate,  using  time  domain  finite  element  analysis.  For  simplicity,  it 
was  assumed  that  the  dynamic  loading  was  fully  in  phase  across  the  plate. 


Introduction 

Aircraft  structures  basically  consist  of  thin,  generally  curved,  plates  attached  to  a  supporting  framework. 
During  flight  these  stiffen^  panels  are  subjected  to  a  combination  of  static  and  dynamic  aerodynamic 
loads.  On  some  aircraft  there  may  be  additional  quasi-static  thermal  loads  due  to  the  impingement  of  jet 
effluxes  in  some  areas.  Parts  of  advanced  short  take  of  and  landing  (ASTOVL)  .aircraft  may  be  required 
to  withstand  noise  levels  up  to  175dB  and  temperatures  up  to  200deg  C.  Under  these  conditions  the 
established  methods  of  dynamic  stress  analysis  for  acoustic  fatigue  design  are  inappropriate  and  cannot 
be  employed. 

British  Aerospace  (BAe),  Sowerby  Research  Centre  (SRC)  and  Military  Aircraft  (MA)  have  been 
developing  a  method  to  predict  the  stress/strain  response  of  aircraft  structures  in  these  extreme  loading 
situations.  The  primary  .consideration  has  been  the  requirement  to  create  an  acoustic  fatigue  design  tool 
for  dealing  with  combined  static  and  dynamic  loads,  including  thermally  generated  “quasi-static”  loads. 


1007 


The  resonant  response  of  thin  aircraft  structures  to  aeroacoustic  loading  is  generally  in  a  firequency 
range  which  implies  that,  if  defects  form,  they  will  quickly  grow.  Hence  to  be  conservative,  it  is 
generally  assumed  that  a  component  has  reached  its  life  when  it  is  possible  to  find  quite  small  defects 
by  non-destructive  evaluation  techniques.  Several  different  materials  and  construction  methods  are  used 
in  modern  aircraft  and  so  there  are  a  number  of  possible  failure  criteria.  In  the  case  of  metals,  it  is  the 
presence  of  cracks  larger  than  a  certain  size.  For  composites  it  can  be  the  occurrence  of  either  cracking 
or  delamination.  Degradation  due  to  the  presence  of  microcracks  may  be  monitored  by  measuring  the 
level  of  stiffness  reduction  which  has  taken  place. 

This  philosophy  simplifies  the  type  of  stress  analysis  needed,  because  it  is  not  necessary  to  model 
structures  with  defects  present.  Materials  can  be  assumed  to  have  simple  elastic  properties  which 
remain  unchanged  throughout  their  lives.  In  consequence,  it  is  necessary  to  know  the  fatigue  behaviour 
in  terms  of  a  direct  relationship  between  number  of  cycles  to  failure  and  the  magnitude  of  the  “nominal” 
cyclic  stress,  or  strain,  at  a  reference  location. 

If  considered  important  and  capable  of  satisfactory  treatment,  the  relationship  can  be  modified  to  take 
into  account  material  property  changes  due  to  the  development  of  very  small  defects  at  points  of  stress 
concentration.  For  example,  metal  plasticity  in  the  region  of  a  small  crack,  could  be  included  in  an 
analysis  of  the  stress  distribution  around  a  fastener  hole.  It  is  well  known  that  plasticity  reduces  the 
peaks  of  stress  which  are  predicted  at  defects  by  analysis  which  assumes  perfectly  elastic  material 
behaviour. 

The  technique  developed  at  BAe  for  modelling  high  acoustic  loads  combined  with  possible  thermal 
buckling  uses  the  time  domain  Monte  Carlo  (TDMC)  technique  together  with  finite  element  analysis  by 
proprietary  FE  codes.  Response  characteristics  are  predicted  directly  in  the  time  domain  using 
simulated  random  acoustic  loadings.  These  may  then  be  used  in  fatigue  life  estimations  which  employ 
cycle  counting  methods  such  as  Rainflow  counting.  It  is  now  practical  to  predict  the  vibrational 
response  of  stiffened  aircraft  panels  without  the  necessity  to  assume  a  linear  response,  and  without 
simplifying  the  spatial  and  temporal  representation  of  the  noise  excitation. 

Since  the  technique  uses  proprietary  finite  element  codes,  quite  large  and  complex  models  of  aircraft 
structure  can  be  analysed  in  a  single  run.  Standard  pre-  and  post-processor  techniques  are  available  to 
speed  up  generation  of  the  finite  element  mesh  and  to  display  the  stress/strain  results. 

The  initial  development  work  was  carried  out  by  modelling  the  random  vibration  of  a  flat  plate.  For 
fully  in-phase  random  loading  at  low  noise  levels  the  predicted  response  is  predominantly  single  mode 
and  at  fhe  frequency  calculated  by  linear  theory.  However,  as  the  decibel  level  is  increased,  the 
frequency  of  the  fundamental  rises  due  to  geometrically  non-linear  stiffening.  At  veiy  high  dB  levels  the 
predicted  response  becomes  multi-modal;  the  resonance  peaks  move  to  higher  and  higher  frequencies 
and  broaden. 

The  effect  of  static  loading  on  the  response  has  been  studied  as  part  of  these  investigations  to  assist  in 
the  validation  of  the  methodology  being  developed. 


Thermal  Effects 

In  some  flight  conditions  it  is  possible  for  a  panel  to  be  buckled  due  to  constrained  thermal  expansion 
and  also  be  subjected  to  very  noise  levels  at  the  same  time.  An  example  is  when  a  ASTOVL  aircraft 
hovers  close  to  the  ground  for  an  extended  period,  panels  which  are  initially  curved,  or  thermally 
buckled  panels  may  possibly  be  snapped  through  from  one  side  to  the  other  by  a  large  increase  in 
dynamic  pressure. 

“Snap-through”  can  be  potentially  damaging  to  the  structure  of  an  aircraft  if  it  occurs  persistently, 
because  the  process  is  associated  with  a  large  change  in  the  cyclic  bending  stress  present  at  the  edge  of 
a  stiffened  panel.  High  performance  aircraft  must  therefore  be  designed  so  that  snap-through  never 
occurs  in  practice. 


1008 


The  dynamic  response  of  curved  panels  or  buckled  flat  plates  is  difficult  to  predict  theoretically  because 
of  non-linear  effects.  The  established  acoustic  fatigue  design  techniques,  which  are  based  on  linear 
vibration  theory,  are  only  able  to  provide  approximate  predictions  of  the  loading  regimes  in  which 
particular  panels  might  be  expected  to  undergo  snap-through. 

The  TDMC  method  can  be  used  to  model  non-linear  multi-modal  vibrations  of  stiffened  aircraft  panels 
which  are  also  subject  to  quasi-static  stress.  In  particular  calculations  may  be  carried  out  in  the  post- 
buckling  regime. 

With  this  technique  simulated  random  dynamic  pressure  loading,  with  measured  or  otherwise  known 
spectral  characteristics  is  applied  to  a  curved,  or  post-buckled  panel  and  the  time  domain  response 
calculated.  The  magnitude  of  the  dynamic  loading  may  then  be  increased  until  persistent  snap-through 
is  observed  in  the  predicted  response.  This  gives  the  designer  the  ability  to  design  out  the  potential 
problem  by  systematically  altering  the  most  important  parameters  in  order  to  identify  the  critical 
regime. 

Fatigue  Life  Estimation 

Although  acoustic  fatigue  is  a  complex  phenomenon,  it  has  been  established  that  the  life  of  a  component 
mainly  depends  on  its  stress/strain  history.  The  most  important  factors  in  this  regard  are  the  magnitude 
and  frequency  of  the  cyclic  strain  and  the  mean  level  of  stress  at  the  likely  failure  points.  On  this  basis 
fatigue  life  can  be  estimated  by  carrying  out  the  three  stage  operation  illustrated  in  Figure  1. 


Stage  1:  Determine  Loads 

A  determination  of  static  design  loads  is  relatively  straightforward  compared  to  a  calculation  of  th.e  full 
temporal  and  spatial  dependence  of  the  aeroacoustic  pressure  on  a  military  jet  in  flight.  This  is  an 
enormous  task  in  computational  fluid  dynamics  (CFD).  Designers  have  to  rely  on  experimental  data 
which  can  come  from  measurements  on  existing  aircraft  or  from  scale  model  tests  of  jets,  for  example. 
Existing  databases  can  be  extrapolated  if  the  circumstances  are  similar.  Experimental  noise  data  is 
usually  in  the  form  of  power  spectral  density  curves  as  opposed  to  time  series  fluctuating  pressures,  but 
either  can  be  used,  depending  on  the  circumstances. 

Stage  2:  Calculate  Stresses 

The  technique  chosen  to  obtain  the  stresses  clearly  depends  on  how  much  knowledge  there  is  about  the 
expected  loads.  In  the  early  stages  of  design  analytic  techniques  would  be  used  to  establish  approximate 
sizes  and  stress  levels.  However,  later  on  when  the  design  is  nearly  completion,  finite  element  (FE) 
stress  analysis  can  be  used  to  model  the  effect  of  random  acoustic  loading  on  the  parts  of  the  skin  which 
are  likely  to  be  severely  affected.  These  calculations  would,  of  course,  be  done  including  the  effect  of 
attached  substructure. 

The  established  method  of  designing  against  acoustic  fatigue  uses  a  frequency  domain  technique  which 
relies  on  the  validity  of  linear  vibration  theory.  The  method  forms  the  basis  of  a  number  of  methodology 
documents  published  by  the  Engineering  Sciences  Data  Unit  (ESDU).  Whenever  there^^e  large  out-of- 
plane  deflections  the  frequency  domain  method  cannot  be  used  because  of  the  “geometric  non¬ 
linearity”.  From  a  strictly  theoretical  point  of  view  such  analyses  have  to  be  carried  out  in  the  time 
domain,  although  approximate  methods  are  applied  with  some  success. 

The  established  technique  produces  inaccurate  results  for  curved  panels,  buckled  panels  and  for  panels 
under  high  amplitude  vibration.  Geometric  non-linearity  usually  stiffens  a  structure  in  bending  so  there 
is  a  tendency  to  overestimate  the  stress  levels  using  the  frequency  domain  technique.  This  conservatism 
is  clearly  useful  from  the  point  of  view  of  safety,  but  it  can  lead  to  possible  “over-design”. 
Unfortunately  this  is  not  always  the  case  when  there  are  compressive  static  stresses  present.  The 
established  method  also  fails  if  the  phase  of  the  noise  varies  significantly  over  the  surface  of  the 
structure,  which  is  the  case  in  a  number  of  aeroacoustic  problems.  The  techniques  under  development 
are  designed  to  overcome  these  problems. 


1009 


Stage  3:  Estimate  Fatigue  Life 


In  cases  of  random  acoustic  loading  it  is  customary  to  assume  that  damage  accumulates  according  to  the 
linear  Miner's  rule.  Fatigue  life  is  determined  from  experimental  data  in  the  form  of  stress  (or  strain 
amplitude),  S,  versus  number  of  cycles  to  failure,  N.  If  a  number  of  cycles,  n,  of  stress/strain,  S,  occur  at 
a  level  of  stress/strain  where  N(S)  cycles  would  cause  failure  then  the  fractional  damage  done  by  the 
n(S)  cycles  is  n(S)/N(S). 

Various  methods  have  been  developed  for  obtaining  n(S)  from  the  stress  (strain)  response.  If  the 
excitation  is  stationary,  ergodic  and  the  response  is  narrow  band  random  then  the  function  n(S)  can  be 
shown  to  be  in  the  form  of  a  Rayleigh  distribution  and  the  damage  sum  can  be  evaluated  from  plots  of 
root  mean  square  stress  (or  strain)  against  number  of  cycles  to  failure.  If  the  statistics  of  the  response 
are  not  Gaussian  then  it  is  necessary  to  count  the  numbers  of  stress  cycles  from  the  time  domain 
response  and  use  constant  amplitude  S/N  curves.  It  is  now  widely  accepted  that  the  best  way  of 
counting  the  cycles  is  to  use  the  Rainflow  method,  [1]. 

The  Loading  Regime 

The  loads  on  an  aircraft  may  be  conveniently  divided  into  static  and  dynamic. 

Loads  which  vary  only  slowly  are: 

a)  Steady  Aerodynamic  Pressure  Loading, 

b)  In-Plane  Loads  transferred  from  “external  structure”,  and 

c)  Thermal  Loads  due  to  Constrained  Expansions. 

The  rapidly  varying  loads  are,  of  course,  the  aeroacoustic  pressure  fluctuations  which  originate  from 
any  form  of  unstable  gas  or  air  flow. 

This  division  is  central  to  the  methodology  which  has  been  developed  because  it  enables  the  modelling 
to  be  carried  out  in  two  distinct  phases.  The,  so  called,  static  loads  do  vary,  of  course,  but  the  idea  is  to 
separate  effects  which  occur  on  a  time  scale  of  seconds  from  the  more  rapidly  varying  acoustic 
phenomena.  The  aim  is  to  split  the  loads  so  that  the  quasi-static  effects  can  be  calculated  in  an  initial 
static  analysis  which  does  not  depend  on  a  particular  dynamic  loading  regime.  Any  aerodynamic 
pressure  may  be  divided  into  a  steady  part  and  a  fluctuating  part.  The  natural  place  to  make  the  cut-off 
is  at  IHz  which  means  that  epoch  times  for  TDMC  simulations  are  then  of  the  order  of  a  second.  The 
epoch  time  must  not  be  too  short  because  of  statistical  errors,  and  it  cannot  be  too  long  because  this 
would  invalidate  the  assumption  of  constant  quasi-static  loads.  In  practice,  there  is  another  constraint  on 
the  epoch  time.  The  number  of  finite  elements  in  the  model  coupled  with  the  premium  on  cpu  time 
places  an  obvious  limit  on  the  epoch  time. 

Comparison  of  the  Time  and  Frequency  Domain  Methods 

A  flow  chart  comparing  the  two  methods  is  given  as  Figure  2.  The  main  difference  between  the  two 
techniques  lies  in  the  representation  of  the  dynamic  loads.  The  FD  method  uses  rms  loadings  and 
spectral  characteristics,  whereas  the  TD  method  uses  the  full  time  series  loadings.  Gaussian  statistics 
are,  de  facto,  assumed  by  the  FD  method,  but  this  is  not  necessarily  the  case  with  the  TD  technique. 

Application  of  the  frequency  domain  method  requires  that  the  response  is  dominated  either  by  a  single 
mode  or  a  small  number  of  modes.  To  determine  whether  or  not  this  is  the  case  in  practice,  a  normal 
modes  analysis  must  be  followed  by  a  determination  of  the  amount  of  coupling  between  the  excitation 
and  each  mode.  This  can  be  determined  quite  accurately  even  if  there  is  a  certain  amount  of  potential 
non-linearity  by  computing  the  joint  acceptances  for  each  mode,  which  are  overlap  integrals  of  the 
mode  shape  functions  with  the  spatial  characteristics  of  the  excitation.  Normally  these  quantities  will  be 
dominated  by  a  few  of  the  low  order  modes.  If  there  is  significant  coupling  into  more  than  one  mode 
then  it  will  be  necessary  to  use  the  TD  method  instead. 

With  the  time  domain  technique  it  is  possible  to  represent  the  dynamic  loads  in  a  way  which  models  the 
convection  of  the  noise  field  across  the  structure.  Very  complicated  loadings  can  be  applied  to  large 


1010 


models  but  in  consequence  it  can  be  difficult  to  validate  the  results  obtained,  because  they  cannot  be 
checked  against  anything  other  than  test  data  which  is  itself  subject  to  confidence  levels.  In  addition  it 
must  be  remembered  that  the  TDMC  results  themselves  are  subject  to  statistical  variability.  Finally  it 
should  be  noted  that  TDMC  data  must  be  used  in  conjunction  with  constant  amplitude  endurance  data. 
Rms  fatigue  data  can  only  be  used  with  frequency  domain  results. 

Time  Domain  Finite  Element  Modelling 

Until  recently,  the  majority  of  finite  element  analyses  were  applied  to  static  loading  conditions  or  “low 
frequency  normal  modes  analysis”.  The  method  involves  the  use  of  an  implicit  code  to  invert  in  one 
operation,  a  single  stiffness  matrix,  which  can  be  very  large.  The  general  availability  of  extremely  fast 
super-computers  has  now  made  it  possible  to  carry  out  large  scale  non-linear  dynamic  finite  element 
modelling  using  explicit  FE  codes.  These  codes  use  very  similar  types  of  element  formulation  to  the 
implicit  ones,  e.g.  shells,  solids  and  bars,  but  the  solution  is  advanced  in  time  using  a  central  difference 
scheme. 

One  potentially  very  useful  capability  of  time  domain  modelling  is  the  application  of  acoustic  pressure 
loadings  which  vary  both  in  the  time  and  spatial  domains.  If  the  spectral  characteristics  are  known, 
either  from  test  or  from  other  modelling  it  is  possible  to  generate  samples  of  random  acoustic  noise  and 
apply  these  directly  to  the  finite  element  model  as  a  series  of  “load  curves”. 

The  technique  for  determining  time  series  noise  was  developed  by  Rice  [2]  and  Shinozuka  [3].  They 
showed  that  homogeneous  Gaussian  random  noise  can  be  generated  from  the  power  spectral  density  as 
a  sum  of  cosine  functions  with  different  frequencies  and  random  phase.  Noise  can  be  temporally  and 
spatially  correlated  noise  by  deriving  phase  differences  from  cross  spectral  functions  if  they  are  known. 

The  TDMC  method  can  be  quite  costly  in  terms  of  central  processor  unit  (cpu)  time  because  the 
solution  must  be  recalculated  at  each  point  in  time.  To  reduce  execution  times,  the  explicit  codes 
employ  reduced  numbers  of  volume  integration  points  in  the  finite  element  formulations.  However  in 
this  work  cpu  times  are  extended  because  long  epoch  times  are  required  to  ensure  adequate  statistics.  It 
can  take  more  than  24hrs  to  obtain  a  solution  over  a  half  second  epoch  if  there  are  a  few  thousand 
elements  in  the  model. 

Hence  there  is  always  a  practical  limit  to  the  size  of  a  particular  time  domain  finite  element  analysis, 
(TDFEA).  If  the  loading  and  geometry  are  not  too  complicated,  the  frequency  domain  method  of 
analysis  can  be  tried  initially  to  gain  more  understanding  of  the  nature  of  the  response  in  an 
approximate  way.  In  some  cases  the  vibrational  response  regime  must  be  considered  carefully  to  decide 
whether  TDFEA  is  really  necessary.  These  may  be  situations  where  the  non-linear  effects  are  only 
moderate. 

It  would  be  ideal  if  the  full  dynamic  response  of  an  aircraft  could  be  determined  with  a  fine  mesh  model 
in  one  huge  operation,  but  experience  has  shown  that  this  requires  too  many  elements.  It  is  possible  to 
construct  frill  models  with  reduced  stiffness  using  superelements,  enabling  flutter  and  buffet  to  be 
studied,  because  these  are  essentially  low  frequency  phenomena.  However,  in  time  domain  analysis  it 
has  been  found  that  models  containing  a  large  amount  of  detail,  such  as  fasteners  and  individual 
composite  material  plys,  require  a  great  deal  of  cpu  time.  To  progress  we  must  devise  some  strategies  to 
overcome  this  situation.  Since  a  full  TDFEA  can  only  handle  a  part  of  the  aircraft  structure,  it  is  very 
important  that  loads  external  to  the  area  under  consideration  are  properly  taken  into  account.  This  is 
cmcial  to  the  success  of  this  type  of  modelling  as  it  is  to  all  finite  element  modelling. 

The  most  important  parameter  in  any  time  series  analysis  is  the  time  step.  This  is  determined  by  the 
velocity  of  sound  waves  in  the  structural  material,  and  is  generally  of  the  order  of  a/v  where  a  is  the 
shortest  element  dimension  and  v  is  the  velocity  of  longitudinal  sound.  A  small  time  step  is  therefore 
required  when  the  elements  are  small  and  the  velocity  of  sound  is  large.  For  an  aluminium  model  with 
10mm  square  elements  the  time  step  is  about  1.6ixs.  Hence  a  TDMC  run  with  a  half  second  epoch  time 
needs  about  a  half  a  million  steps.  A  simple  5000  shell  element  calculation  on  a  Cray  C94  would  take 
approximately  10  hours. 


1011 


Dynamic  FE  models  of  aircraft  structure  can  be  constructed  in  many  ways,  using  shell  elements,  beam 
elements  and/or  solid  elements.  Special  elements  exist  for  damping  and  for  sliding  interfaces.  Joints  can 
be  modelled  with  sliding  interfaces,  or  with  short  beams,  or  just  with  tied  nodes.  Fasteners  can  be 
modelled  with  small  solid  elements,  with  short  beams  or  with  tied  nodes,  also.  Unfortunately,  however 
short  beams  and  small  solid  elements  cause  a  dramatic  lowering  of  thC  time  step.  For  example,  if  the 
smallest  fastener  dimension  is,  say  3mm,  the  time  step  will  have  to  be  reduced  to  about  O.Sfis  if  solid 
elements  are  used  in  the  model.  The  effect  on  cpu  time  is  such  as  to  make  the  calculations  impractical. 
Sliding  interfaces  are  an  efficient  way  to  model  skin/substructure  contact  in  explicit  analyses,  but  it  is 
important  to  choose  the  algorithm  carefully  because  some  techniques  can  consume  large  amounts  of  cpu 
time. 

The  best  practical  way  of  representing  stiffened  aircraft  panels  for  TDMC  analyses  is  considered  to  be 
with  four  noded  shell  elements  simply  tied  together  at  their  edges.  A  number  of  efficient  shell 
formulations  are  avail-able  and  meshes  can  be  rapidly  produced  from  the  design  geometry.  Of  course, 
such  models  cannot  be  expected  to  produce  highly  accurate  stress  data  in  the  region  of  small  features 
but  this  aspect  has  to  be  sacrificed  in  the  interests  of  achieving  statistically  significant  amounts  of  time 
series  data.  To  improve  the  accuracy  of  stress  predictions  in  the  neighbourhood  of  stiffeners  etc.,  it  will 
be  necessary  to  couple  TDMC  analyses  with  fine  mesh  static  analyses. 


The  Generation  of  Time  Series  Data 

A  number  of  factors  must  be  borne  in  mind  when  generating  time  series  data  for  TDMC  calculations.  It 
is  important  to  consider  carefully  the  frequency  range  and  number  of  points  which  define  the  load 
spectrum  in  conjunction  with  the  epoch  time  and  number  of  points  on  the  time  series. 

The  Nyquist  Criterion  [4]  states  that  the  time  increment  must  be  less  than  or  equal  to  one  over  twice  the 
upper  frequency  on  the  power  spectral  density  curve.  For  the  sake  of  argument,  take  the  upper 
frequency  to  be  IkHz.  This  means  that  the  time  increment  must  be  less  than  500ps.  A  more 
conservative  time  increment  is  based  on  the  requirement  to  represent  the  dynamic  response  of  the 
structure  as  accurately  as  possible  over  a  full  cycle.  Assuming  a  resonant  frequency  of  500Hz,  which  is 
perhaps  near  the  limit  in  practice,  and  10  points  per  cycle  which  is  more  than  sufficient,  the  lower  limit 
on  the  time  step  works  out  at  about  2C)0|is.  Taking  all  these  factors  into  consideration,  the  number  of 
points  on  the  spectrum  curve  should  be  of  the  order  of  1000  and  there  should  be  between  1000  and 
5000  on  the  time  series.  Longer  epoch  times  can  be  used  but  for  reasons  of  practicality  and  statistics  it 
is  better  to  run  more  than  one  short  epoch  simulation  rather  than  one  long  simulation. 

Explicit  FE  modelling  frequently  requires  that  the  time  step  be  smaller  than  200jj,s.  In  the  example 
given  above  the  time  step  required  by  the  explicit  code  was  1.6fis.  Under  these  circumstances  the 
random  noise  could  be  defined  with  a  smaller  time  increment,  but  going  to  this  level  of  effort  has  been 
found  to  produce  no  measurable  change  to  the  calculated  response. 

Static  Initialisation 

There  are  two  possible  ways  of  dealing  with  the  effect  of  static  loads  in  TDMC  modelling.  Firstly  the 
complete  analysis  can  be  carried  out  using  the  explicit  code.  To  do  this  it  is  necessary  to  apply  only  the 
static  loads  to  the  model  and  run  the  code  until  equilibrium  is  reached.  By  introducing  a  high  level  of 
artificial  damping  the  stresses  created  can  be  relaxed  in  a  relatively  short  period  of  time.  The  time 
required  depends  on  the  lowest  resonant  frequency  of  the  structure  and  the  size  of  the  smallest  element 
in  the  model.  This  facility  is  termed  “dynamic  relaxation”. 

The  alternative  is  to  make  use  of  another  facility  in  the  explicit  code  called  “static  initialisation”.  The 
deformed  shape  and  stress  state  of  the  structure  with  just  the  static  loads  applied  are  first  obtained  very 
quickly  using  an  implicit  code.  The  solution  for  the  stressed  state  is  then  initialised  into  the  explicit 
code  prior  to  the  application  of  the  dynamic  loads.  Dynamic  relaxation  may  be  used  to  smooth  out  any 
differences  between  the  models. 


1012 


Damping  Representation 

Vibrating  aircraft  structures  are  damped  by  several  mechanisms,  for  example  friction  at  joints,  re¬ 
radiation  of  acoustic  waves,  and  energy  loss  in  viscoelastic  materials  It  is  difficult  to  generalise  about 
the  relative  importance  of  each  damping  process  in  practice.  Also  reliable  quantitative  data  is  not 
available  in  sufficient  detail  to  justify  the  inclusion  of  complex  models  of  damping  into  the  TDMC 
analyses.  Test  results  on  vibrating  stiffened  aluminium  panels  tend  to  show  that  the  damping  is  best 
approximated  by  a  combination  of  mass  and  stiffness  proportional  coefficients.  There  is  a  range  of 
frequencies  in  which  the  damping  ratio  can  be  considered  to  be  roughly  constant.  Until  more  detailed 
experimental  data  are  available  the  most  expedient  approach  is  to  assume  a  nominal  value  for  the  global 
damping  ratio  which  does  not  change  with  frequency.  Over  the  years  it  has  become  standard  practice  to 
assume  a  damping  ratio  of  about  2%  for  fastened  aluminium  structures. 

Equivalent  Linearisation 

There  are  some  loading  regimes  in  which  the  non-linear  response  to  high  levels  of  random  acoustic 
loading  can  be  approximately  found  using  a  linearisation  technique  combined  with  a  frequency  domain 
analysis.  The  basic  idea  is  to  replace  the  non-linear  stiffness  term  in  the  general  vibration  equation  by  a 
linear  term  such  that  the  difference  between  the  rms  response  of  the  two  equations  is  minimised  with 
respect  to  a  shifted  fundamental  resonance  frequency.  If  an  approximate  equation  for  the  non-linear 
stiffness  is  known  then  it  is  possible  to  derive  an  expression  for  the  shifted  “non-linear”  resonance 
frequency.  The  rms  response  to  random  acoustic  loading  may  then  be  found  by  combining  the 
Miles/Clarkson  equation  with  some  form  of  static  geometrically  non-linear  analysis.  References  to  this 
technique  are  Blevins  [5],  Mei  [6]  and  Roberts  &  Spanos  [7].  Where  the  geometry  is  complex  the  most 
appropriate  form  of  analysis  is  clearly  finite  element  analysis. 

Implementation  and  Validation  Studies 

The  stress  analysis  work  described  in  this  paper  has  been  undertaken  using  MSC-NASTRAN  and 
LLNL-DYNA. 

NASTRAN  is  a  well  known  implicit  finite  element  code  which  is  capable  of  handling  very  large 
numbers  of  elements.  It  has  been  developed  very  much  with  aerospace  structural  analysis  in  mind.  It  is 
basically  a  linear  analysis  code,  although  there  are  a  large  number  of  adaptations  to  deal  with  non-linear 
problems.  It  can  also  function  as  a  dynamic  code,  but  is  much  slower  than  DYNA  in  this  mode  because 
it  basically  needs  to  solve  the  complete  problem  at  each  time  step.  The  non-linear  features  which  are 
most  relevant  to  the  type  of  stress  analysis  being  discussed  here  are  those  concerned  with  geometric 
non-linearity.  Geometric  non-linearity  is  treated  by  dividing  the  load  into  a  series  of  steps,  obtaining  the 
solution  incrementally.  In  this  work,  the  code  has  been  used  for  linear  and  non-linear  static  analyses  and 
for  normal  modes  analyses. 

DYNA  is  an  explicit  finite  element  code  originally  developed  for  the  calculation  of  the  non-linear 
transient  response  of  three  dimensional  structures.  The  code  has  shell,  beam  and  solid  element  models 
and  there  are  a  large  number  of  non-linear  and/or  anisotropic  material  models  available.  DYTMA  was 
developed  primarily  for  the  modelling  of  impact  and  there  is  no  limit,  as  far  as  the  code  is  concerned, 
on  the  size  of  finite  element  model  which  can  be  analysed.  Many  of  its  advanced  features  relate  to 
impact  modelling  and  are  not  required  for  this  work.  One  useful  feature,  however  is  the  laminated 
composite  material  model  based  on  the  equivalent  single  layer  approximation.  This  code  has  been  used 
for  the  TDMC  calculations  presented  in  this  paper. 

Finite  element  models  for  the  stress  analyses  were  produced  using  MSC-PATRAN,  It  has  a  wide  range 
of  geometry  and  mesh  generation  tools  and  now  has  built  in  interfaces  for  both  NASTRAN  and  DYNA. 
The  element  definitions  are  compatible  with  both  codes  and  it  is  a  simple  matter  to  toggle  between  the 
two  codes  by  changing  the  analysis  preference.  Not  all  the  features  of  DYNA  are  supported  and  some 
of  the  parameters  must  be  set  by  editing  the  DYNA  bulk  data  produced  by  PATRAN. 

NASTRAN  results  were  post-processed  using  PATRAN.  DYNA  results  were  post  processed  using 
TAURUS,  which  is  faster  and  easier  to  use  than  PATRAN  for  this  task;  Some  special  in-house  codes 
have  been  written  to  generate  random  acoustic  noise  from  power  spectral  densities,  as  described  above, 


1013 


and  to  post  process  time  series  output  from  TAURUS.  One  of  the  codes  incorporates  a  fast  Fourier 
transform  (FTT)  routine  to  determine  spectral  responses  from  the  DYNA  time  series  predictions.  These 
codes  are  covered  under  the  generic  title  “NEW-DYNAMIC”. 

TDMC  Calculations  on  a  Simple  Flat  Plate 

Calculations  have  been  performed  on  a  very  simple  model  to  implement  the  TDMC  technique  and 
develop  the  in-house  software  referred  to  above.  A  PATRAN  database  was  constructed  representing  a 
simple  flat  plate,  350mm  x  280mm  x  1.2mm  thick  as  an  array  of  shell  elements  34x28.  For  simplicity 
the  boundary  conditions  were  taken  either  as  simply  supported  or  clamped.  There  are  a  number  of 
alternative  shell  element  formulations  available  in  DYNA,  [8].  The  Hughes-Liu  shell  was  used  initially 
because  of  its  good  reputation  for  accuracy,  but  later  a  switch  was  made  to  a  similar,  but  slightly  faster 
shell  element,  called  the  YASE.  It  was  found  that  equally  satisfactory  results  could  be  obtained  more 
quickly  using  this  element. 

Analyses  without  Static  Loads 

A  series  of  DYNA  calculations  were  carried  out  with  a  fiiUy  correlated  random  acoustic  pressure  load 
with  a  flat  noise  spectrum  between  OHz  and  1024Hz.  Investigations  were  carried  out  into  the  effect  of 
varying  the  sound  pressure  level,  the  epoch  time,  the  mesh  resolution,  the  damping  coefficient  and  the 
stochastic  function. 

Figure  3  shows  the  displacement  response  of  the  central  node  of  the  model  for  a  sound  pressure  level  of 
115dB  (about  12Pa  rms),  simply  supported  edges  and  mass  proportional  damping  set  so  that  the 
damping  ratio  was  equal  to  2%  at  the  fundamental  (1,1)  resonance  of  the  plate.  The  corresponding 
spectral  response  is  shown  in  Figure  4.  shows  a  sharp  resonance  peak  at  a  frequency  of  61.0Hz  which  is 
very  close  to  the  theoretical  frequency  of  the  (1,1)  mode  for  the  simply  supported  plate.  The  in-phase 
loading  means  that  only  the  modes  with  odd  numbered  indices  are  excited.  I^e  peaks  corresponding  to 
the  (3,1)  and  (1,3)  modes  are,  however,  not  visible  on  the  plot  because  they  are  too  small.  It  may  be 
concluded  from  these  results  that  the  behaviour  of  the  plate  at  these  pressure  level  is  well  within  the 
linear  regime. 

An  investigation  into  the  behaviour  of  the  rms  displacement  response  as  a  fijnction  of  SPL  was  carried 
out  by  increasing  the  loading  incrementally  from  75dB  (0.1 2Pa  rms)  to  175dB  (12kPa  rms).  The  results 
are  shown  in  Figure  5.  Also  shown  are  theoretical  predictions  obtained  using  the  Miles/Clarkson 
formula  with  NASTRAN  linear  and  non-linear  analyses  as  explained  above,  see  below  for  discussion. 

The  statistical  variation  of  the  results  was  investigated  by  repeating  a  half  second  epoch  TDMC  run  ten 
times  with  different  samples  of  flat  spectrum  noise.  It  was  found  that  the  standard  error  of  the  rms 
response  was  about  16%.  A  second  set  of  ten  repeats  were  carried  out  with  the  epoch  increased  to  2.5s. 
In  this  case  the  standard  error  reduced  to  roughly  8%.  From  the  theory  of  stochastic  processes,  it  can  be 
shown  that  the  standard  error  is  inversely  proportional  to  the  square  root  of  the  epoch  time.  On  this 
basis  therefore  the  ratio  between  the  standard  errors  should  be  equal  to  the  square  root  of  five,  or  2.23. 
From  the  analyses  this  ratio  is  about  2.  Further  runs  established  that  these  results  are  not  affected  by  the 
vibration  amplitude,  even  when  the  response  becomes  non-linear. 

Cautiously  therefore,  it  can  be  concluded  that  the  variance  of  the  TDMC  results  is  unaffected  by  non¬ 
linearity  of  the  response.  This  is  an  important  finding  because  it  builds  confidence  in  the  technique.  In 
many  practical  situations  it  may  be  necessary  to  rely  on  just  one  simulation  and  an  appropriate  factor  of 
safety.  It  can  be  quite  time  consuming  to  carry  out  a  large  number  of  repeat  TDMC  simulations.  The 
level  of  variance  would  be  first  established  by  repeating  one  load  case  a  number  of  times,  before 
confidently  applying  it  to  the  results  of  other  load  cases. 

Comparisons  with  Linearised  Theory 

The  linear  theory  of  plate  bending,  [9],  leads  to  relationships  between  the  central  deflection,  w,  of  a 
rectangular  plate  and  a  uniform  static  pressure  load,  Pstat  which  take  the  following  form. 

Psutab  =  kcffW  (1) 


1014 


where  a  and  b  are  the  length  and  breadth  of  the  plate,  and  k^ff  is  an  effective  stiffness  parameter  which  is 
a  function  of  the  modulus  of  rigidity  of  the  plate  and  the  edge  boundary  conditions.  For  the  plate 
studied  k^ff  is  about  30N/mm  for  the  case  of  simply  supported  edges  and  lOON/mm  for  clamped  edges. 
The  above  equation  only  holds,  however,  at  very  low  amplitudes,  as  can  be  seen  from  Figure  6.  This 
compares  geometrically  non-linear  NASTRAN  predictions  with  the  linear  ^eory.  Curves  are  shown  for 
both  simply  supported  and  clamped  boundary  conditions.  The  finite  element  results  show  the 
characteristic  hardening  spring  type  of  non-linearity. 

At  higher  amplitudes  the  dynamic  behaviour  may  be  approximately  predicted  using  “equivalent 
linearisation”  theory,  which  assumes  that  the  response  remains  predominantly  single  mode,  but  with  a 
resonant  frequency  which  rises  as  the  stiffness  of  the  structure  increases.  When  the  deflection  is  large 
the  static  force-deflection  relationship  can  be  written  as  the  sum  of  a  linear  stiffness  term  and  a  cubic 
non-linear  term: 

psiatab  =  kw(l+pw^)  (2) 

where  a  b  is  the  force,  k  is  the  linear  stiffness.  The  equation  is  written  with  the  leading  term  factored 
out  to  emphasis  the  point  that  p.  is  a  constant  which  is  small  compared  to  the  rms  deflection.  In  the  limit 
of  small  w  we  can  expect  the  pw^  term  in  the  brackets  to  be  negligible  compared  to  one,  which  means 
that  the  k  in  this  equation  must  be  the  same  as  kcff  above. 

Equation  2  was  fitted  to  the  NASTRAN  results  shown  in  Figure  6  to  find  the  best  fit  values  of  k  and  p. 
Table  1  shows  the  results  compared  with  the  effective  stiffness  calculated  from  linear  plate  bending 
theory.  It  can  be  seen  that  the  theoretical  stiffness  is  almost  identical  to  the  best  fit  k  from  the  non-linear 
finite  element  analysis. 

With  reference  to  the  results  in  Figure  5,  it  is  obvious  that  the  nature  of  the  response  is  strongly 
dependent  on  the  amplitude  of  the  vibrations.  For  rms  displacements  up  to  about  4%  of  the  plate 
thickness  the  behaviour  was  completely  linear.  For  displacements  between  4%  and  150%  of  plate 
thickness,  the  response  was  essentially  single  mode  dominated  but  the  level  could  not  be  predicted  by 
the  Miles/Clarkson  approach.  The  “equivalently  linear”  solution  does,  however,  agree  with  the  DYNA 
result  up  to  a  displacement  of  about  l.8mm.  The  linearisation  approach  cannot  be  expected  to  be 
correct  for  displacements  above  about  1.5  times  plate  thickness.  Above  this  point  the  response  predicted 
by  DYNA  was  multi-modal  and  strongly  non-linear.  The  equivalently  linear  predictions  departed 
considerably  from  the  DYNA  results  when  the  vibration  amplitude  was  very  high. 

It  was  also  observed  that  the  frequency  response  peaks  became  increasingly  noisy  for  higher  pressures, 
representing  the  increased  level  of  non-linearity  in  the  plate  vibrations.  The  increase  in  the  frequency  of 
the  fundamental  mode  with  acoustic  pressure,  as  calculated  by  DYNA,  is  shown  in  Figure  7.  Predictions 
from  equivalent  linearisation  theory  and  from  the  theory  of  Duffing's  equation  are  also  included,  see 
Nayfeh  &  Mook  [10]. 


t„,  =  f(l  +  3nw™,Y’ 

Equiv.  Lin.  Pred. 

(3) 

f„,  =  f(l+(3/8)w„,^) 

Duffing's  Eq.  Pred. 

(4) 

The  DYNA  results  lie  mostly  between  the  two  theoretical  curves,  agreeing  particularly  well  with  the 
results  of  equivalent  linearisation  theory  up  to  around  700Pa  (151dB  rms).  The  level  of  agreernent 
obtained  shows  that  the  frequency  response  behaviour  of  the  DYNA  model  is  similar  the  theoretical 
predictions,  providing  an  independent  check  on  the  results.  As  might  be  expected,  at  around  700Pa  the 
agreement  begins  to  breakdown,  since  the  linearisation  theories  are  not  valid  for  deflections  which  are 
significantly  greater  than  the  plate  thickness.  It  may  be  concluded,  however,  that  the  effect  of  geometric 
non-linearity  at  high  amplitudes  is  being  computed  by  DYNA  in  a  reasonably  accurate  manner.  A 
detailed  comparison  with  experimental  data  is  needed  to  determine  the  accuracy  of  the  DYNA  response 
predictions  themselves. 


1015 


Due  to  the  increasingly  irregular  shape  of  the  frequency  response  functions  derived  from  the  DYNA 
time  series  predictions  at  high  acoustic  loads,  it  was  not  possible  to  calculate  very  accurate  peak  widths 
for  pressures  above  approximately  135dB  (I20Pa  rmsX  Figure  8  shows  that  the  width  of  the  peak 
increased  with  increasing  acoustic  pressure,  but  not  in  a  regular  manner.  When  the  damping  is  mass 
proportional,  equivalently  linear  theory  predicts  that  the  width  of  the  peak  should  remain  unchanged  as 
the  pressure  rises.  This  is  because  the  geometric  stiffening  effect  of  rising  acoustic  pressure  exactly 
cancels  the  effect  of  a  smaller  damping  ratio  at  the  higher  resonant  frequency.  This  graph  shows  this  as 
a  horizontal  straight  line  at  2.44Hz.  The  DYNA  result  is  closer  to  the  type  of  behaviour  observed 
experimentally  where  the  width  of  the  peak  generally  increases  with  increasing  the  sound  pressure  level. 

Analyses  with  Combined  Loads 

Further  work  was  conducted  with  static  loads  superimposed  on  different  levels  of  random  acoustic 
loading.  These  calculations  were  done  using  the  coupled  NASTRAN-DYNA  approach  outlined  earlier. 
That  is  to  say  the  deformed  geometry  was  obtained  by  applying  the  static  loads  to  a  NASTRAN  model, 
with  the  results  being  initialised  into  DYNA  and  dynamically  relaxed  before  the  dynamic  loading  was 
applied.  Calculations  were  performed  with  compressive  in-plane  loading,  static  pressure  loading  and 
thermal  loading.  With  the  exception  of  the  thermal  runs,  the  boundary  conditions  used  in  these  runs 
were  identical  to  clamped,  except  that  symmetrical  in-plane  movement  of  the  edges  was  permitted.  We 
have  called  these  conditions  “semi-clamped”.  It  has  been  found  that  the  fundamental  resonant  frequency 
of  the  plate  without  static  loading  is  only  reduced  by  a  very  small  amount  if  the  appropriate  in-plane 
degrees  of  freedom  are  released,  see  Figure  9.  These  boundary  conditions  are  actually  closer  to  those 
which  exist  in  reality  when  a  panel  in  built  into  a  larger  structure. 

Figures  10-14  show  results  of  some  of  the  analyses  which  have  been  carried  out.  They  give  time  series 
data  along  with  spectra  responses  calculated  by  the  in-house  post-processing  code.  Numerical  data 
derived  from  these  results  are  summarised  in  Tables  3-5. 

A  series  of  analyses  have  been  carried  out  with  compressive  in-plane  loads  equal  to  one  third  of  the 
theoretical  buckling  loads  in  compression.  For  the  plate  used,  the  forces  per  unit  side  length  were  - 
3.46N/mm  in  the  x-direction  and  -5.46N/mm  in  the  y-direction.  The  results  of  one  analysis  are  shown  in 
Figure  10.  It  has  been  found  that  the  response  remains  dominated  by  the  fundamental  (1,1)  mode  as 
long  as  the  plate  is  unbuckled  and  the  SPL  is  low.  The  softening  effect  of  the  compressive  loads  on  the 
frequency  agreed  quite  well  with  Rayleigh-Ritz  predictions,  [5],  up  to  an  SPL  of  ?dB.  At  higher  sound 
pressure  levels,  the  DYNA  results  reflected  stiffness  changes  which  were  greater  than  those  predicted 
by  the  theory.  The-  same  was  found  in  the  case  of  tensile  loading.  It  is  believed  that  these  differences  are 
due  to  approximations  built  into  both  the  Rayleigh-Ritz  theory  and  the  DYNA  code. 

Figure  1 1  shows  the  results  of  a  calculation  with  a  superimposed  normal  pressure.  The  magnitude  of  the 
pressure,  700Pa,  was  chosen  so  as  to  provide  an  example  of  “post-buckled”  analysis.  This  size  of 
pressure  causes  the  plate  to  bow  out  in  the  centre  by  about  0.6mm.  It  is  well  known  that  in  the  post- 
buckling  regime  the  random  response  of  a  plate  depends  upon  the  magnitudes  of  both  the  static  and 
dynamic  loads.  In  this  case  the  static  loading  was  large  compared  to  the  applied  dynamic  loads  and 
“snap-through”  did  not  occur.  The  plate  simply  vibrated  about  its  statically  deflected  position  in  the 
fundamental  mode  with  a  slightly  increased  frequency. 

To  provide  a  test  of  the  DYNA  thermal  stressing  capability,  and  to  carry  out  an  investigation  into 
“snap-through”,  several  analyses  were  carried  out  with  a  uniform  temperature  rise  of  lOdeg  C  applied  to 
the  plate  with  clamped  edges.  This  is  quite  sufficient  to  cause  buckling  because  the  resulting 
compressive  biaxial  stress,  c,  is  well  above  the  buckling  level,  Gb-  If  f  is  the  frequency  of  the 
fundamental  and  J  is  a  constant  equal  to  1.248  because  of  the  clamped  boundary  condition,  the  two 
stresses  can  be  determined  approximately  from 


a  =  EaT/(l-v^) 

(5) 

Cb  =  4pa^f2/J 

(6) 

1016 


where  E,  a,  v  and  p  are  Young's  modulus,  coefficient  of  thermal  expansion,  Poisson's  ratio,  and  density 
respectively.  Using  these  formulae  we  find  c  =  24MPa  and  Ob  =  14MPa. 

Analyses  were  carried  out  with  several  different  levels  of  dynamic  load.  The  results  of  three  of  the 
calculations  are  shown  in  Figures  12,13  and  14.  It  was  found  that  the  threshold  for  snap-through 
occurred  at  an  acoustic  load  of  about  IkPa,  see  Table  4.  Below  this  level  the  mean  deflection,  w,  is  a 
function  of  the  static  load  alone,  equal  to  about  2.8nun  (the  negative  sign  indicates  that  the  plate  has 
bowed  in  direction  of  negative  z).  At  higher  SPLs  the  mean  deflection  reduces  because  the  plate  snaps 
backwards  and  forwards  between  positive  and  negative  z.  The  calculated  response  spectra  for  these 
higher  level  runs,  show  an  additional  peak  at  a  very  low  frequency,  ie  less  than  lOHz.  This  is  an  artifact 
caused  by  the  snap-through  since  the  fundamental  resonance  of  the  clamped  plate  is  at  1 13Hz. 

Figures  10-14  all  show  probability  density  functions  derived  from  the  time  series  data.  The  fluctuations 
on  these  plots  are  caused  by  the  smallness  of  the  epoch  time.  In  all  cases,  except  for  the  thermal 
calculations  with  the  two  largest  acoustic  loads,  it  can  be  seen  that  the  functions  are  basically  Gaussian 
in  shape.  It  may  therefore  be  concluded  that  it  is  reasonable  to  assume  that  the  response  of  a  plate  in  the 
post-buckled  region  is  Gaussian  unless  there  is  a  large  amount  of  snap-through. 

Discussion 

The  work  described  is  the  starting  point  for  investigations  and  validations  using  more  complex  FE 
models.  Further  work  has  been  carried  out  using  models  including  curvature,  sub-structure  and  detailed 
features.  It  is  difficult  to  validate  the  predictions  obtained  from  such  models  by  comparing  with  test  data 
because  the  results  themselves  are  open  to  interpretation.  It  has  been  found  that  the  predicted  stress 
levels  are  closer  to  the  test  results  when  the  chosen  location  is  away  from  any  small  features.  The  lack 
of  good  agreement  in  the  neighbourhood  of  the  features  can  be  explained  by  the  relatively  coarse  mesh 
used  in  the  dynamic  models.  The  overall  level  agreement  was  much  better  than  that  between  predictions 
based  on  linear  or  equivalently  linear  theory  and  test.  On  the  basis  of  experience,  the  latter  tend  to  over¬ 
predict  by  upwards  of  factors  of  two  and  three.  From  this  work  it  has  been  found  that  the  DYNA 
predictions  tend  to  be  greater  than  test  by  amounts  which  vary  but  are  generally  much  less.  The  average 
over-prediction  was  about  40%  with  a  significant  change  as  a  function  of  location. 

TDMC  runs  can  take  a  significant  amount  of  computer  time  to  carry  out  and  it  is  believed  that  to  make 
further  improvements  the  technique  should  be  combined  with  detailed  stressing  using  static  finite 
element  analysis.  Inaccurate  results  can  be  obtained  if  the  boundaries  of  the  part  of  the  structure  under 
analysis  are  not  properly  restrained.  In  the  case  of  models  of  aircraft  panels  this  may  significantly  affect 
the  resonant  frequencies  which  in  turn  affects  the  level  of  calculated  dynamic  stress.  In-plane  loads  on  a 
panel,  perhaps  due  to  thermal  stressing,  can  alter  the  fundamental  by  as  much  as  100-200Hz.  Looked  at 
from  a  theoretical  point  of  view,  the  only  way  to  solve  this  problem  is  to  construct  a  second,  coarse 
model  of  the  component,  along  with  some  of  its  surrounding  structure.  An  initial  calculation  can  then  be 
carried  out  with  this  model  in  order  to  obtain  the  loads  and  boundary  conditions  for  subsequent 
application  to  the  original  model. 

The  dynamic  phenomenon  of  “snap-through”  cannot  be  modelled  using  existing  methods  and  so  the 
TDMC  /  finite  element  technique  offers  the  engineer  a  way  to  determine  where  the  likely  regions  of 
unstable  vibration  are  located  in  circumstances  where  the  structure  is  complicated  by  attachments  etc. 


Conclusions 

This  paper  has  sought  to  explain  how  time  domain  finite  element  modelling  can  be  used  to  assist  in  the 
design  of  aircraft  against  acoustic  fatigue.  Although  the  technique  is  computationally  intensive,  it  does 
have  a  place  in  the  effort  to  understand  complex  vibrations,  such  as  the  response  of  structures  to 
spatially  correlated  jet  noise  excitations,  or  interactions  between  high  sound  pressure  levels  and  thermal 
loads. 

The  work  at  BAe  is  continuing  in  an  attempt  to  provide  the  analyst  with  a  greater  ability  to  determine 
dynamic  stress  levels  in  advanced  structures  with  complex  loadings. 


1017 


References 


1.  Dowling  N.  E.  ,  Fatigue  Prediction  for  Complicated  Stress  Strain  Histories,  J  Materials  1,  71 
(1972). 


2.  Rice,  In  Selected  Papers  on  Noise  and  Stochastic  Processes,  Ed  N  Wax  pplSO,  Dover  New  York 
(1954). 

3.  Shinozuka  M.  ,  Computers  and  Structures,  2,  855,  (1972). 

4.  Bendat  J.  S.  and  Piersol  A.G.  ,  Engineering  Applications  of  Correlation  and  Spectral  Analysis 
Wiley  (1990). 

5.  Blevins  R.D,  ,  An  approximate  method  for  sonic  fatigue  analysis  of  plates  and  shells,  J  Sound  and 
Vibration,  129,  1,  51  (1989). 

6.  Mei  C.  and  Paul  D.B.  ,  Non  Linear  multi-modal  response  of  a  clamped  rectangular  plates  to 
Acoustic  Loading,  AIAA  Journal,  24,  634,  (1986). 

7.  Roberts  J.  B.  &  Spanos  P.D  ,  Random  Vibration  and  Statistical  Linearisation,  Wiley,  (1990). 

8.  Whirley  R.G.  and  Engelmann  B.E.  ,  DYNA3D:  A  Nonlinear,  Explicit,  Three  Dimensional  Finite 
Element  Code  for  Solid  and  Structural  Mechanics  —  User  Manual,  Lawrence  Livermore  National 
Laboratory,  UCRL-MA- 107254  Rev.  1,  (Nov  1993). 

9.  Szilard  R. ,  Theory  and  Analysis  of  Plates,  Prentice  Hall,  New  Jersey. 

10.  Nayfeh  A.H.  &  Mook  D.T.  ,  Non  Linear  Oscillations,  Wiley  (1979). 


1018 


Parameter 

lUQgnnggll 

HISSH 

Linear  Theory 

NASTRAN 

NASTRAN 

Simply  Supported 

30.0 

30.7 

1.09 

Clamped 

104. 

101. 

0.266 

Table  1:  Values  of  Parameters  fitted  to  NASTRAN  results  compared  with  the 
linear  theory  values. 


Static  Loads 

Acoustic  Load 
Rrms 
(Pa/dB) 

Theory  ] 

1  DYNA 

(N.mm 

N, 

(N.mm 

Freq 

(Hu) 

Wfim 

(mm) 

■SI 

Wrms 

(mm) 

None 

1.2  (95.6) 

jnggi 

0.00290 

114 

0.01.2 

-3.46 

1.2  (95.6) 

0.00316 

94.1 

0.00313 

-3.46 

643.5  (150.1) 

1.69 

115 

1.08 

-3.46 

1.2  (95.6) 

68.33 

0.00359 

68.1 

0.00368 

-3.46 

700  (150.9) 

68.33 

2.09 

103 

1.42 

Table  2:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  compressive  in-plane  loads. 


Static  Load 
p(Pa) 

Acoustic  Load 
(Pa/dB) 

DYNA 

None 

1.2  (95.6) 

114 

0.0000 

0.01.2 

700 

None 

— 

-0.646 

— 

700 

12  (115.6) 

115 

-0.655 

0.0189 

700 

700  (150.9) 

134 

-0.477 

1.28 

5  k 

None 

— 

-2.89 

— 

5  k 

12  (115.6) 

179 

-2.93 

0.0152 

5  k 

700  (150.9) 

173 

-2.70 

0.999 

Table  3:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  static  pressure  loads. 


Temperature 
T  (<»C) 

Acoustic  Load 
(Pa  /  dB) 

1  DYNA  1 

None 

1.2  (95.6) 

114 

0.00256 

10 

None 

— 

-2.85 

— 

10 

1.2  (95.6) 

234 

-2.86 

0.0140 

10 

700  (150.9) 

219 

-2.80 

0.197 

10 

Ik  (154.0) 

9.01 

2.47 

1.02 

10 

1.2k  (155.6) 

5.01 

-0.717 

2.24 

10 

1.5k  (157.5) 

3.00 

-0,376 

2.20 

10 

2k  (160,0) 

92.2 

0.0185 

1.95 

10 

4k  (166.0) 

195 

0.0568 

2.08 

Table  4:  Summary  of  results  of  calculations  with  random  acoustic  loading 
superimposed  on  a  thermal  load. 


1019 


Figure  2:  Flowchart  Illustrating  the  Frequency  and  Time  Domain  Techniques 


(uiUi),ueujeoB,ds,a  ^  O 


Displacement  of  central  node  for  12Pa  rms  acoustic  pressure, 

1. 


DYNA 


Figure  4:  Spectral  Response  of  the  fiat  plate  corresponding  to  Fig  3.  Central 
Node  for  12Pa  rms  pressure,  DYNA  calculation. 


3 


2.5 


2 

1 

o 

^  1 


0.5 


0 

Figure  5:  Rms  Central  Deflection  of  the  plate  versus  sound  pressure  level 
Comparison  between  DYNA  results  and  linear  theory. 


3 


2.5 


2 

■£ 

£ 

E 

I 

I 

o 

1 


0.5 


0 

Figure  6:  Central  deflection  of  the  plate  versus  pressure,  NASTRAN  calcula¬ 
tions  compared  to  linear  theory. 


500  1000  1500  2000  2500  3000  3500 

Static  Pressure  (Pa) 


rms  Pressure  (Pa) 


1022 


Figure  7:  Variation  of  fundamental  frequency  of  the  (1,1)  mode  with  rms 
pressure,  Comparison  between  DYNA  and  theory. 


Damping  2%;  Stoctiastic  Fn  #1 ;  3Sx29  Nodes 

0  - 1 - 1 - 1 - 1 - - 

0  200  400  600  800  1000  1200 

rms  Pressure  (Pa) 

130  140  145  150  153  155 

SPL  (dB) 

Figure  8:  Variation  of  width  of  the  (1,1)  mode  resonance  peak  with  rms  pres¬ 
sure,  Comparison  between  DYNA  and  theory. 


1023 


Figure  9:  DYNA  model  predictions  for  random  vibration  of  the  plate  with 
semi-clamped  boundary  conditions. 


Figure  10:  Random  Vibration  results  with  compressive  load  in  the  y-direction 
of  -3.46N/mm  and  SPL  of  l.OPa. 


Figure  11:  Random  Acoustic  Loading  of  SPL=12Pa  superimposed  on  a  static 
pressure  of  TOOPa. 


1024 


Figure  12:  Random  acoustic  loading  of  SPL=700Pa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  —  No  Snap  Through 


Figure  13:  Random  acoustic  loading  of  SPL— 2kPa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  — Nearly  continuous  snap-through. 


Figure  14:  Random  acoustic  loading  of  SPL=4kPa  superimposed  on  a  thermal 
load  of  10  deg,  clamped  edges.  — Dominant  acoustic  load. 


1026 


SYSTEM  IDENTIFICATION  II 


ROBUST  SUBSYSTEM  ESTIMATION 
USING  ARMA-MODELLING 
IN  THE  FREQUENCY  DOMAIN 

by  U.  Prells,  A.  W.  Lees,  M.  1.  Friswell  and  M.  G.  Smart, 

Department  of  Mechanical  Engineering  of  the  University  of  Wales 

Swansea, 

Singleton  Park,  Swansea  SA2  8PP,  United  Kingdom 

ABSTRACT 

This  paper  reflects  early  results  of  the  research  on  modelling  the  influence  of 
the  foundation  on  the  dynamics  of  the  rotor.  The  foundation  is  connected  to 
the  rotor  via  journal  bearings.  Dynamic  models  exist  for  the  subsystems  of 
the  rotor  and  of  the  bearings;  the  first  is  reliable  but  the  latter  is  uncertain. 
The  foundation  model  is  unknown  and  has  to  be  estimated  using  rundown 
data.These  are  measured  responses  of  the  foundation  at  the  bearings  due 
to  unbalance  forces  of  the  rotor  which  are  assumed  to  be  known.  Uncer¬ 
tainties  in  the  bearing  model  will  be  transfered  to  the  estimated  foundation 
parameters.  The  main  scope  of  this  paper  is  to  introduce  a  method  which 
enables  the  decoupling  of  the  problem  of  model  estimation  and  the  problem 
of  the  influence  of  the  bearing  model  uncertainty. 

The  influence  of  changes  in  the  model  of  the  bearings  on  the  estimation 
of  the  foundation  model  is  mainly  due  to  the  sensitivity  of  the  computed 
forces  applied  to  the  foundation  at  the  bearings.  These  are  used  together 
with  the  associated  measured  responses  to  estimate  the  foundation  model  in 
the  frequency  domain.  Using  an  ARMA  model  in  the  frequency  domain  it  is 
possible  to  estimate  a  filtered  foundation  model  rather  than  the  foundation 
model  itself.  The  filter  is  defined  in  such  a  way  that  the  resulting  force  has 
minimum  sensitivity  with  respect  to  deviations  in  the  model  of  the  bearings. 
This  leads  to  a  robust  estimation  of  the  filtered  model  of  the  foundation. 
Since  the  filter  can  be  defined  in  terms  of  the  models  of  the  rotor  and  of  the 
bearings  only,  the  problems  of  estimating  the  foundation’s  influence  and  of 
the  sensitivity  of  the  estimates  with  respect  to  the  model  of  the  bearings 
are  decoupled. 

The  method  is  demonstrated  by  a  simple  example  of  a  single-shaft  rotor. 
Even  if  the  errors  in  the  bearing  model  are  about  50  %  the  relative  input  and 
output  errors  of  the  filtered  foundation  model  are  of  the  same  magnitude 
as  the  round-off  and  truncating  errors. 

1  INTRODUCTION 

An  important  part  of  a  machine  monitoring  system  for  fault  diagnostics  of 


1027 


a  turbo  generator  is  a  reliable  mathematical  model.  This  model  includes 
the  subsystems  of  the  rotor,  the  bearings  and  the  foundation.  The  model  of 
the  rotor  represents  the  most  reliable  knowledge,  the  model  of  the  journal 
bearings  is  uncertain,  and  despite  of  intensive  research  it  is  not  yet  possible 
to  define  a  model  for  the  foundation  which  refiects  the  dynamical  contribu¬ 
tion  to  the  rotor  with  sufficient  accuracy.  The  first  step  to  determine  the 
contribution  of  the  foundation  on  the  rotor’s  dynamic  performance  is  the 
estimation  of  a  reliable  foundation  model. 

Rundown  data  are  available,  i.e.  displacements  ufb{<->^)  ^  of  the 
foundation  at  the  bearings  which  are  due  to  an  unbalance  force  /c/(a;)  G 
of  the  m-shaft  rotor,  given  at  discrete  frequencies  a;  €  :=  {wi,  ■  •  • ,  wm}? 

and  this  data  may  be  used  to  estimate  the  foundation  model.  A  com¬ 
mon  method  ([1],[2],[3],[4],[5])  is  to  estimate  the  unknown  dynamic  stiffness 
matrix  6  i^4mx4m  foundation  at  the  bearings  using  the  in¬ 

put/output  equation 

F{uj)upb{(^)  =  (1) 

where  the  force  fpB  of  the  foundation  at  the  bearings  can  be  expressed 
by  dynamic  condensation  in  terms  of  the  data  ups,  fu  and  in  terms  of  the 
dynamic  stiffness  matrices  Ar,B  of  the  rotor  and  the  bearings  respectively 
yielding 

fpB  =  -Bufb  +  [0,  B]A~^q  ^  ^  .  (2) 

Here  the  dynamic  stiffness  matrix  Arb  of  the  rotor  mounted  on  the  bearings 
is  partitioned  with  respect  to  the  n  inner  degrees-of-freedom  (dof)  of  the 
rotor  and  to  the  4m  connecting  (interface)  dof 


^RB 


Aru  Arib 
Arbi  Arbb  +  B 


(3) 


The  non-zero  components  of  the  force  fpi  G  (D”  in  eq.  (2)  of  the  inner  part 
of  the  rotor  are  the  components  of  the  unbalance  force  /y,  i.e.  introducing 
the  control  matrix  Su  €  ^  dynamic  stiffness 

matrix  Ar  of  the  rotor  is  given  in  terms  of  the  matrices  of  inertia  and 
stiffness  which  are  defined  by  modal  analysis  and  by  its  physical  data  given 
by  the  manufacturer.  Each  of  the  m  shafts  of  the  rotor  is  connected  to 
the  foundation  usually  via  2  journal  bearings.  Since  the  dynamic  stiffness 
matrix  B  of  the  journal  bearings  represents  a  model  for  the  oil  film  it  consists 
of  connecting  dof  only.  It  can  be  shown  that  B  is  block  diagonal 


B  = 


0 


0 


(4) 


containing  the  dynamic  stiffness  matrices  Bi  =  Ki+  jujDi.i  =  1,  •  •  •  ,m,  of 
the  m  bearings.  Ki,  Di  are  the  matrices  of  stiffness  and  damping  respectively 


1028 


which  result  from  linearisation  and  are  in  general  non-symmetric  and  non¬ 
singular.  Eq.  (1)  is  then  used  to  estimate  the  foundation  transfer  function 
F{u).  This  has  been  discussed  in  several  papers  ([1],[2],[3],[4],[5]).  Lees  et 
al.  [3]  pointed  out  that  fpB  is  sensitive  with  respect  to  deviations  in  the 
model  of  the  bearings  over  part  of  the  frequency  range.  This  sensitivity  is 
transfered  to  the  model  estimates. 

In  this  paper  a  method  is  introduced  which  enables  the  decoupling  of  the 
two  problems  of  model  estimation  and  of  sensitivity  of  the  foundation  model 
with  respect  to  the  model  of  the  bearings.  The  basic  idea  of  this  method  is 
to  estimate  a  transfer  function  H{u)  which  maps  the  displacements  ufb{^) 
to  a  force  /^(w)  rather  than  the  force  i-e. 

H(u)ufb{^)  =  (5) 

In  extension  of  the  earlier  method  the  force  //f(a;)  can  be  chosen  to  be  of 
minimum  sensitivity  with  respect  to  the  model  of  the  bearings.  This  robust 
estimated  transfer  function  H{uj)  is  related  to  that  of  the  foundation  F(uj) 
by  a  transformation  P{u) 


H{lo)  =  PMF(a;),  (6) 

which  of  course  retains  the  sensitivity  with  respect  to  the  model  of  the 
bearings.  But  since  P{uj)  only  depends  on  the  models  of  the  rotor  and  the 
bearings  in  the  case  of  a  modification  within  the  model  of  the  bearings  no 
new  model  estimation  has  to  be  performed  because  this  has  been  done  ro¬ 
bustly  with  respect  to  such  model  changes. 


2  THE  OPTIMUM  CHOICE  OF  THE 
FORCE  VECTOR 

As  stated  in  Lees  et  al.  [3]  the  sensitivity  of  the  force  /fb  with  respect  to  B 
is  mainly  due  to  the  inversion  of  the  matrix  Arb  in  eq.  (2).  It  can  be  shown 
that  the  condensation  method  of  estimating  the  force  /fb  results  from  the 
special  case  of  eliminating  the  last  4m  rows  of  the  matrix 


W  := 


Arii 

^RBI 

0 


Arib 

ArbB  -b  P 


^  ^(n+8m.)x(n+4m) 


(7) 


which  can  be  written  as  Arb  =  T'^W  ^  (l^(n+4m.)x(n+4m)  defining  the 
selecting  matrix  of  the  master  dof  as 

T  :=  [ei,  •  •  •  ,  €  R(-+8n)x(n+4m)^  (S) 


1029 


where  in  general  en  denotes  a  unit  vector  of  appropriate  dimension  contain¬ 
ing  zeros  everywhere  but  in  the  nth  place.  In  extension  to  the  force  fpB 
defined  in  eq.  (2)  for  an  arbitrary  selecting  matrix  T  €  the 

condensation  leads  to  a  force  fn  given  by 


j-^LT 


In+Sm-W{T'^Wy'T^ 

”  *■  V*'  . . . 

=:P 


( 

Bufb 

V  } 


(  ^  \ 
0 

V  -^4m  J 


fpB- 


(9) 


Here  T-^  €  denotes  the  matrix  which  selects  the  slave  dof, 

and  in  general  In  denotes  the  unit  matrix  of  dimension  n.  Indeed,  inserting 
the  special  choice  of  T  from  eq.  (8)  into  eq.  (9)  leads  to  the  sensitive  force 
Ih  =  fpB  as  defined  in  eq,  (2). 

The  reason  for  the  sensitivity  of  /fb  is  that  the  subsystem  of  the  rotor 
has  low  damping.  Near  the  resonance  frequencies  of  Arb  its  large  condition 
number  depends  sensitively  on  B.  Thus  the  sensitivity  of  Jfb  with  respect 
to  B  is  due  to  a  large  condition  number  of  Arb  •  Let  T  denote  the  set  of  all 
possible  selecting  matrices,  i.e. 

r  :=  {[ei..--',ew4„]:e4elR"+*“. 

l<4<n  +  8m,  V  /c  =  1,  •  • « ,n  + 4m},  (10) 

One  criterion  for  an  optimum  choice  of  the  force  Jr  may  be  formulated  as 
the  following  minimisation  problem: 

Criterion  1: 

The  optimum  choice  is  the  solution  of 

mmcond{T^W),  (11) 

where  W  is  defined  in  eg.  (7). 

A  low  condition  number  is  necessary  but  not  sufficient  in  order  to  provide 
a  low  sensitivity  of  the  force  /h*  Therefore  a  numerical  test  can  be  applied 
using  stochastic  deviations  in  the  bearing  model.  Let  A  A  consist  of 

uniform  distributed  non-correlated  random  numbers  with  zero  mean  values 
and  variances  equal  to  1/3  for  alH  G  1,  ■  •  * ,  2m.  Define 

ABi  =  ABfisi,  Ti)  :=  siAKi  -h  A  A,  (12) 

where  the  positive  scalars  si.Ti  control  the  magnitude  of  the  random  error 
of  the  i-th  bearing  model.  Thus,  the  error  AB  =  AB{s,r)  of  the  bear¬ 
ing  model  is  well  defined  for  s  :=  (si,  •  ■  • ,  52771)"^  and  r  :=  (ri,  •  •  •  ,r2mV ■ 


1030 


Regard  the  force  fn  =  f{uj,T,AB)  as  a  function  of  the  selecting  ma¬ 
trix  T  and  the  bearing  model  error  AB.  For  I  random  samples  AB{k)  = 
AB(r{k),s{k)),k  ==  calculate  for  each  frequency  a;  G  the  up¬ 

per  and  lower  bounds  for  the  real  and  imaginary  part  of  each  component 
/i,  ^  =  1,  •  •  • ,  4m  of  the  force  vector  /,  i.e.: 


//Lax(“.U 

;=  max  Re  {/i(a;,T,  AB(/c))}  , 

(13) 

:=  min  Re  {fi{u,T,  AB{k))}  , 

(14) 

:=  ^max^Im  {/i(a;,T,  AB(/c))}  , 

(15) 

fLni^.T) 

:=  min  Im  AB{k))}  . 

k=l,"-,l 

(16) 

Defining  the  force  vectors 

/max(w,T) 

■■=  T)+j-  T), 

(17) 

/mm(w,T) 

(18) 

the  second  criterion  can  be  formulated  as  a  minimax  problem: 

Criterion  2: 

The  optimum  selection  is  obtained  from 

4m 

minmax^l/imax(w,r)  -  (19) 

TST  uj£u  .  - 
2  =  1 

Before  the  method  outlined  is  demonstrated  by  an  example  some  aspects  of 
the  mathematical  model  of  the  foundation  and  methods  for  its  estimation 
based  on  the  input/output  equation  (5)  will  now  be  considered. 


3  ESTIMATION  OF  THE  FOUNDATION 

MODEL 


The  purpose  of  this  section  is  to  estimate  the  unknown  foundation  model 
represented  by  the  matrix 

F{uj)  =  Afbb{^)  -  Afbi{^)Af]j{uj)Afib{^)‘  (20) 


This  expression  results  from  dynamic  condensation  of  the  dynamic  stiffness 
matrix  of  the  foundation 


Af 


Afbb  Afib 
Afib  Afh 


(21) 


1031 


which  is  partitioned  with  respect  to  its  inner  dof  (index  I)  and  those  dof 
coupled  to  the  bearings  (index  B).  For  viscous  damped  linear  elastomechan- 
ical  models  the  dynamic  stiffness  matrix  Aj?(a;)  of  the  foundation  is  given 
by  ^ 

Af(uj)  :=  (22) 

i=0 

The  matrices  Ai  are  real  valued  and  represent  the  contributions  of  stijffness, 
damping  and  inertia  for  z  =  0, 1, 2  respectively.  In  this  case  the  identifica¬ 
tion  of  the  foundation  model  requires  the  estimation  of  the  three  matrices 
Ai  which  are  parameterised  by  introducing  dimensionless  adjustment  pa¬ 
rameters  aik  €  IR,  for  all  /c  =  1,  *  •  • ,  ^  =  0, 1, 2  (see  for  instance  [6]  or 

[7] ).  Those  parameters  are  related  to  given  real- valued  matrices  Sik  by 

Ni 

Ai{ai) 

Jk=i 

Writing  the  adjustment  parameters  as  one  vector  a"’’  ;=  (af,aj’,aj)  G  IR^, 
p  :=  ATq  +  Ni  +  W2,  the  estimation  of  the  foundation  model  is  equivalent  to 
the  estimation  of  the  parameter  vector  a.  The  dynamic  stiffness  matrix  of 
the  foundation  becomes  a  nonlinear  function  of  this  parameter  vector 

^"(0;)  =F[(jj^a)  —  Afbb[^-)0)  “  Afb7(ci;j a)Ap}j(a;,  a)Af’/5(a;,  a).  (24) 

Substituting  the  measured  quantities  for  upB  and  fu  into  eqs.  (5)  and 
(9)  the  parameter  vector  a  is  usually  estimated  by  minimising  some  norm  of 
the  difference  between  measured  and  calculated  quantities,  called  residuals 

[8] .  Using  equation  (5)  is  equivalent  to  the  input  residual  method.  Defining 
the  ith  partial  input  residual  as 

(25) 

where  the  dependency  on  the  model  parameters  a  of  the  input  vector  is 
defined  by 

f{u),  a)  :=  P{uj)F[u),  a)uFB(w),  (26) 

the  cost  function  to  be  minimised  is  given  by 

M 

JKa):=£^;}(i)W,(i)^/(i),  (27) 

i=l 

where  Wj{i)  represents  a  weighting  matrix  for  the  zth  partial  residual  and 
the  superscript  f  denotes  the  conjugate- transpose.  The  inverse  problem  (27) 
is  nonlinear  with  respect  to  the  parameters  to  be  estimated.  Thus,  there 
is  no  advantage  relative  to  the  output  residual  method.  Defining  the  ith 
partial  output  residual  as 

vo{i)  :=  u{ui,  a)  -  (28) 


1032 


where  the  model  output  is  defined  by 


(29) 


0 

0 

ApiiiuJ^a)  . 
(30) 


(31) 

Woii)  denotes  a  weighting  matrix  for  the  ith  partial  output  residual. 

Mathematical  modelling  is  always  purpose  orientated  [9].  In  the  case 
discussed  in  this  paper  the  purpose  is  to  estimate  the  influence  of  the  foun¬ 
dation  on  the  dynamics  of  the  rotor.  For  this  purpose,  no  physically  inter¬ 
pretable  model  is  necessary  in  order  to  model  this  influence.  In  the  next 
section  an  alternative  mathematical  model  is  introduced  which  leads  to  a 
linear  inverse  problem. 

3.1  ARMA  MODELLING  IN  THE  FREQUENCY 
DOMAIN:  THE  FILTER  MODEL 

Auto  Regressive  Moving  Average  models  are  well  developed  (see  for  in¬ 
stance  [10], [7])  in  order  to  simulate  dynamic  system  behaviour.  ARMA 
models  are  defined  in  the  time  domain  by 

-  lAt)  =  -  ^A^),  (32) 

i=0  i=0 

where  the  present  output  (state  or  displacement)  u{t)  due  to  the  present 
input  f{t)  depends  on  rio  past  outputs  and  on  rii  past  inputs. 

In  the  frequency  domain  eq.  (32)  leads  to  a  (frequency-)  filter  model  [7]. 
With  reference  to  eq.  (5)  it  has  the  form 


Su 

0 

0 

0 


u(w,a)  :=  [0,0,/2(m+i),0].4  ^(w,a) 
with  the  dynamic  stiffness  matrix  A  of  the  entire  model 
A{u,  a)  = 


Arii{u)  Arib{^^)  0 

Arbi{^)  Arbb{‘^) B{lo)  —B(u) 

0  —B{uj)  R(a;)  4- Afbs(^)  g) 

0  0  Afib{^jO,) 


the  cost  function  to  be  minimised  is 


M 


Jo(a)  :=E^oW^oW«o(i). 


The  output  and  input  powers  Uo^rii  respectively,  and  the  matrices 
(•SA:)fc=o,-,ni  are  called  filter^parameters  and  Jiave  to  be  esti¬ 
mated.  Of  course  the  minimum  of  det[A(a;)]  and  of  det[B(cj)]  correspond 
to  the  resonance  and  anti-resonance  frequencies  of  the  subsystem  of  the 
foundation  respectively. 

For  an  optimum  choice  of  P  (see  eq.  (9))  the  estimation  of  A  and  B  can 
be  considered  to  be  independent  of  the  precise  values  of  the  model  of  the 
bearings.  Thus,  the  problem  of  the  uncertainty  in  the  bearmg  models  and 
the  problem  of  model  estimation  are  decoupled.  If  A  and  B  are  estimated 
refering  to  eq.  (5)  then 

=  H.  (34) 

The  estimation  of  the  filter  parameters  is  robust  with  respect  to  deviations 
in  the  bearings  model.  Thus,  the  uncertainty  of  the  estimation  of  the  foun¬ 
dation  model  F  is  due  to  the  inversion  of  the  matrix  P  only 

F  =  P-^H  =  p-'-B~^A,  (35) 

which  represents  a  problem  a  priori  and  which  occurs  only  in  the  calculation 
of  the  force  of  the  foundation  at  the  bearings 

/PB  =  P-^B~^Aufb-  (36) 

Of  course  the  force  vector  fpB  is  sensitive  to  changes  in  the  bearings  model 
but  only  due  to  corresponding  changes  in  P.  The  estimated  part  B  A  is 
robust  with  respect  to  changes  in  the  bearings  model. 

In  order  to  calculate  the  response  ufb  iio  explicit  calculation  of  the 
inverse  of  P  is  necessary, 

upB  =  A  Bfn-  (37) 

Since  the  estimated  model  and  the  force  Jh  are  insensitive  with  respect  to 
the  bearings  model  the  estimation  otupB  is  robust  in  this  sense. 

Of  course  the  influence  of  errors  in  up b  and  fu  have  not  yet  been  taken 
into  account.  Accordingly  the  model  powers  n^n^,  must  be  estimated  as 
well  as  the  matrices  Ai.Bk-  The  estimation  method  is  outlined  in  the  fol¬ 
lowing  section. 

3.2  ESTIMATION  OF  THE  FILTER 
PARAMETERS 

In  order  to  estimate  the  filter  parameters  the  least  squares  method  can  be 
applied  to  minimise  the  equation  error  in  eq.  (33).  Defining  the  zth  partial 
equation  residual  as 

vsij)  '=  A[ui)upB[^i)  -  B{uJi)fH{^i)  (38) 


1034 


the  cost  function  to  be  minimized  is  given  by 


M 

i=l 


(39) 


where  W^ii)  denotes  a  weighting  matrix  for  the  ith.  equation  residual.  As¬ 
suming  WeIi)  =  Am  for  alH  =  1,  •  •  ’ ,  M,  the  filter  equation  (33)  can  be 
extended  for  M  excitation  frequencies  as 

=  (40) 


where  C/,  2^  and  A  are  defined  by 


U 

Z 

A 


•  •  •  5  , 

r  a;i  0  1 


lom 


(41) 

(42) 

(43) 


The  solution  of  the  minimisation  problem  (39)  is  equivalent  to  the  normal 
solution  of  eq.  (40)  which  can  be  rewritten  as 


[Afio  1 ' ' '  1  -Ao )  ‘  :  ■^o] 


=:  V 


U 

Z 


=  0 


(44) 


Because  the  filter  parameters  represented  by  the  matrix  V  € 
j^4mx4m(no+ni+2)  leal-valued,  equation  (44)  must  be  satisfied  for  the 

real  and  imaginary  parts  of  the  matrix  Y  €  which  finally 

yields 

1/ [Re  {Y}  ,  Im  {Y}]  =:  VA  =  0.  (45) 

This  problem  does  not  lead  to  a  unique  solution  for  the  filter  parameters. 
Indeed,  for  any  arbitrary  non-singular  matrix  C 

CAufb  =  (46) 


is  also  a  solution.  But  since  one  is  interested  (see  eq.  (34))  in  the  product 
B~^  A  (or  its  inverse)  only  this  final  result  is  of  interest  and  this  product  is 
unique. 


1035 


As  a  necessary  and  a  sufficient  condition  for  a  full-rank  solution  V  of 
eq.  (45)  the  matrix  X  €  5^4m(no+ni4-2)x2M  a  rank  deficiency  of 

4m,  i.e. 

rank(A)  =  4m  (no  +  n^  +  1).  (47) 

Of  course  this  problem  has  to  be  treated  numerically.  The  rank  decision 
is  usually  made  by  looking  to  the  singular  values  7(no,  rii)  6  5^4m(no+ni+2) 
of  the  matrix  X  =  X{no,ni).  Because  one  cannot  expect  to  achieve  zero 
rather  than  relative  small  singular  values  one  has  to  define  a  cut-off  limit. 
This  is  due  to  the  fact  that  the  equation  error  (39)  can  be  made  arbitrary 
small  by  increasing  the  degree  p  :=  no  +  Ui  of  the  filter  model.  The  same 
situation  occurs  if  one  looks  at  the  maximum  relative  input  error 


6/  := 


max 


(48) 


or  to  the  maximum  relative  output  error 


en  :==  max 


II^fs(^2)1I 


(49) 


With  increasing  degree  p  the  errors  ej  and  cq  can  be  made  arbitrary  small. 

This  is  a  typical  expression  for  an  ill-posed  problem  which  can  be  turned 
into  an  well-posed  problem  by  applying  regularisation  methods  [11].  To 
choose  an  appropriate  regularisation  method  needs  further  investigation  and 
is  beyond  the  scope  of  this  paper.  In  the  next  section  the  method  of  choosing 
the  optimum  force  vector  fn  is  demonstrated  by  a  simple  example. 


4  A  SIMPLE  EXAMPLE 

The  test  model  is  depicted  in  Fig.  1.  The  one-shaft-rotor  is  simulated  by 
an  Euler-Bernoulli  beam  which  is  spatially  discretised  with  10  dof.  Accord¬ 
ing  to  the  partition  with  respect  to  inner  points  and  interface  points  (see 
eq.  (3))  the  number  n  of  inner  rotor  dof  is  8  and  the  number  of  connecting 
dof  is  2.  Only  one  translation  dof  of  the  rotor  is  connected  to  each  bearing 
which  are  modelled  by  massless  springs  with  stiffnesses  ki  —  1.77  •  10®  and 
k2  =  3-54  •  10®  N/m  respectively.  The  foundation  is  modelled  by  an  uncon¬ 
nected  pair  of  masses  mi  =  90,  m2  =  135  kg  and  springs  with  stiffnesses 
kfi  =  /c/2  =  1.77  •  10®  N/m.  The  force  fu  due  to  an  unbalance  6  =  0.01 
kg-m  is  given  by  fu{<^)  ■=  €  IR.  The  force  vector  f^i  e  IR®  is  assumed 

to  have  one  non-zero  component  only,  i.e.  Jri  fuc^.  The  frequency  range 
between  0  and  250  Hz  is  discretised  with  equally  spaced  stepsize  of  0.5  Hz. 
The  selecting  matrix  T  €  of  the  master  dof  is  assumed  to  consist  of 

the  unit  vector  64  in  order  to  select  out  the  unbalance  force  fjj  because  this 


1036 


Figure  1:  The  simple  test  model 

excitation  is  independent  of  the  model  of  the  bearings  and  therefore  of  min¬ 
imum  sensitivity.  Thus  the  remaining  redundancy  consists  in  eliminating 
one  row  of  the  matrix 

G  :=  [ei,  82. 83. 65.  •  •  • .  eul''  W  €  (50) 

The  result  of  the  first  criterion  are  depicted  in  Fig.  2.  It  shows  the  frequency 
dependent  condition  numbers  for  the  elimination  of  each  row  of  the  matrix 
G  in  turn.  This  leads  to  an  optimum  choice  by  eliminating  the  4-th  row  of 
G.  Thus,  the  optimum  choice  of  the  master  dof  is  given  by  T*^  =  [64,65]. 
This  result  is  confirmed  by  applying  the  second  optimisation  criterion. 

For  this  purpose  a  uniform  distributed  uncorrelated  random  error  with 
zero  mean  value  is  added  to  the  stiffnesses  of  the  bearings  simulating  a 
model  variation  of  50  per  cent,  i.e.  ki  — ^  ki  -1-  Nzikil2,  where  = 

1,2,  are  uncorrelated  random  numbers  with  expectation  value  E{Aki}  =  0 
and  with  variance  E{AkiAkj}  =  For  a  size  of  I  -  500  random 

samples,  the  upper  and  lower  bounds  /max, /min  and  /f  max, /h min  of  the 
force  fpB  =  /fb(<^,  A5)  and  of  the  force  Jh  =  /(w,T,  AB)  respectively 
with  the  selecting  matrix  T  -  [64,65],  have  been  calculated.  In  contrast  to 
the  maximum  difference  of  upper  and  lower  bounds  of  the  force  f^B  of  ~  20, 
that  of  the  force  fn  is  of  the  order  of  the  computational  accuracy  ~  10"'^, 
and  is  therefore  negligible.  In  a  first  step  the  force  /fb  is  used  to  estimate 
the  filter  parameters  of  the  model  F  of  the  foundation.  Solving  the  singular 
value  decomposition  for  all  input  and  output  powers  {ni.rio)  6  [0,  5]^  C 
the  calculation  of  the  maximum  relative  input  and  of  the  maximum  output 
errors  as  defined  by  eqs.  (48)  and  (49)  with  fn  ~  Ifb  leads  to  the  results 


1037 


Elimination  of  row  number... 


Frequency  [Hz] 


Frequency  [Hz] 

Figure  2:  Frequency  dependent  condition  numbers 

depicted  in  Fig.  3.  For  a  model  realisation  with  [ni.rio)  =  (0,2)  the  values 
of  the  maximum  relative  errors  are  approximately  ej  7.1  •  and  eo  ~ 
9.0-10“”^  which  corresponds  to  the  computational  accuracy.  Using  this  model 
the  estimates  of  the  filter  matrices  correspond  within  the  computational 
accuracy  to  those  of  the  ‘true’  foundation  model. 

Using  variations  of  the  force  fpB  between  the  bounds  /max,  /min  froni 
the  second  criterion  the  associated  upper  and  lower  bounds  of  the  relative 
input  and  output  error  have  been  calculated.  The  influence  of  the  variation 


1038 


Firgure  3:  Maximum  relative  input  and  output  error  for  different 
input  powers  rii  and  output  powers  rio  using  the  force  vector  fpB 


of  the  bearing  stiffness  of  50%  leads  via  the  associated  variation  of  the 
force  fpB  to  drastic  variations  of  the  relative  input  and  output  errors.  The 
difference  of  upper  and  lower  bound  of  the  maximum  relative  input  error  is 


1039 


Output  power 


Figure  4:  Maximum  relative  input  and  output  error  for  different 
input  powers  rii  and  output  powers  Uo  using  the  optimised  force 

vector  Jh 

of  order  100  and  that  of  the  maximum  relative  output  error  is  approximately 
LOl 


The  situation  is  different  using  the  optimised  force  fn  in  order  to  es¬ 
timate  the  filter  parameters  of  the  model  H.  For  each  input  and  output 
power  {ni,no)  e  [0,25]^  C  the  maximum  relative  input  error  and  the 
maximum  relative  output  error  have  been  calculated.  The  result  is  shown  in 
Fig.  4.  For  a  maximum  relative  input  error  e;  6.5  •  10“®  a  filter  model  of 
degree  12  is  available  with  the  powers  (n^,  =  (4, 8).  This  model  produces 

a  maximum  relative  output  error  eo  ^  10“®. 

Analogous  to  the  robustness  investigations  for  fpB  now  for  the  force 
/h  the  upper  and  lower  bounds  of  the  relative  input  and  of  the  relative 
output  error  due  to  the  random  variation  in  the  bearing  models  have  been 
calculated.  For  the  chosen  model  with  powers  (nj,no)  =  (4, 8)  the  difference 
of  upper  and  lower  bound  of  the  relative  input  error  as  well  that  of  the 
relative  output  error  are  of  about  the  same  order  10“^.  Thus,  compared 
with  the  order  of  variation  100  and  10"^  of  the  direct  foundation  model 
estimate  the  estimate  of  H  is  robust  with  respect  to  changes  in  the  bearing 
model. 


5  CONCLUSION 

In  this  paper  a  method  is  introduced  which  enables  the  decoupling  of  the 
two  problems  of  model  estimation  and  of  sensitivity  of  the  foundation  model 
with  respect  to  the  model  of  the  bearings.  The  method  produces  an  opti¬ 
mised  choice  of  the  input /output  equation  which  provides  a  transfer  func¬ 
tion  estimation  that  is  robust  with  respect  to  deviations  in  the  model  of 
the  bearings.  For  the  foundation  model  estimation  a  filter  model  is  intro¬ 
duced.  This  modelling  strategy  has  the  advantage  of  leading  to  a  linear 
inverse  problem.  The  disadvantage  is  that  with  increasing  model  degree 
the  equation  error  can  be  made  arbitrarily  small.  Because  this  error  should 
not  become  smaller  than  the  accuracy  of  the  data,  a  cut-off  limit  has  to  be 
determined  a  priori.  Further  investigations  should  allow  the  cut-off  limit  to 
be  related  to  the  data  errors. 

REFERENCES 

1.  Feng,  N.S.  and  Hahn,  E.J.,  Including  Foundation  Effects  on  the  Vi¬ 
bration  Behaviour  of  Rotating  Machinery.  Mechanical  Systems  and 
Signal  Processing,  1995  Vol.  9,  No.  3,  pp.  243-256. 

2.  Friswell,  M.I.,  Lees,  A.W.  and  Smart,  M.G.,  Model  Updating  Tech¬ 
niques  Applied  to  Turbo-Generators  Mounted  on  Flexible  Founda¬ 
tions.  NAFEMS  Second  International  Conference:  Structural  Dynam¬ 
ics  Modelling  Test,  Analysis  and  Correlation,  Glasgow:  NAFEMS, 
1996  pp.  461-472. 


1041 


3.  Lees,  A.W.  and  Friswell,  M.L,  Estimation  of  Forces  Exerted  on  Ma¬ 
chine  Foundations.  Identification  in  Engineering  Systems,  Wiltshire; 
The  Cromwell  Press  Ltd.,  1996,  pp.  793-803. 

4.  Smart,  M.G,,  Friswell,  M.L,  Lees,  A.W.  and  Prells,  U.,  Errors  in 
estimating  turbo-generator  foundation  parameters.  In  Proceedings 
ISMA21  -  Noise  and  Vibration  Engineering,  ed.  P.  Sas,  Katholieke 
Universiteit  Leuven,  Belgium,  1996,  Vol.  II,  pp.  1225-1235 

5.  Zanetta,  G.A.,  Identification  Methods  in  the  Dynamics  of  Turbogener¬ 
ator  Rotors.  The  International  Conference  on  Vibrations  in  Rotating 
Machinery,  IMechE,  C432/092,  1992,  pp.  173-181. 

6.  Friswell,  M.L  and  Mottershead,  J.E.,  Finite  Element  Model  Updating 
in  Structural  Dynamics.  Dordrecht,  Boston,  London:  Kluwer  Aca¬ 
demic  Publishers,  1995. 

7.  Natke,  H.G.,  Einfuhrung  in  die  Theorie  und  Praxis  der  Zeitreihen- 
und  Modalanalyse  -  Identifikation  schwingungsfdhiger  elastomechanis- 
cher  Systeme.  Braunschweig,  Wiesbaden:  Friedrich  Vieweg  Sz  Sohn, 
1993. 

8.  Natke,  H.G.,  Lallement,  G.,  Cottin,  N.  and  Prells,  U.,  Properties  of 
Various  Residuals  within  Updating  of  Mathematical  Models.  Inverse 
Problems  in  Engineering,  Vol.  1,  1995,  pp.  329-348. 

9.  Natke,  H.G.,  What  is  a  true  mathematical  model?  -  A  discussion  of 
system  and  model  definitions.  Inverse  Problems  in  Engineering,  1995, 
Vol.  1,  pp.  267-272. 

10.  Gawronski,  W.  and  Natke,  H.G.,  On  ARMA  Models  for  Vibrating 
Systems.  Probabilistic  Engineering  Mechanics,  1986,  Vol.  1,  No.  3, 
pp.  150-156. 

11.  Baumeister,J.,  Stable  Solution  of  Inverse  Problems.  Braunschweig, 
Wiesbaden:  Friedrich  Vieweg  &;  Sohn,  1987. 

ACKNOWLEDGEMENTS 

The  authors  are  indebted  to  Nuclear  Electric  Ltd  and  Magnox  Pic  for  fund¬ 
ing  the  research  project  BB/G/40068/A  to  develop  methods  which  enables 
the  estimation  of  the  influence  of  the  foundation  on  the  dynamics  of  the 
rotor.  Dr.  Friswell  gratefully  acknowledges  the  support  of  the  Engineering 
and  Physical  Sciences  Research  Council  through  the  award  of  an  Advanced 
Fellowship. 


1042 


MATHEMATICAL  HYSTERESIS  MODELS  AND 
THEIR  APPLICATION  TO  NONLINEAR 
ISOLATION  SYSTEMS 


Y.Q.  Ni,  J.M.  Ko  and  C.W.  Wong 

Department  of  Civil  and  Structural  Engineering 
The  Hong  Kong  Polytechnic  University,  Hong,  Kong 

Abstract 

Two  mathematical  hysteresis  models,  the  Duhem-Madelung  (DM) 
model  and  the  Preisach  model,  are  introduced  to  represent  the 
hysteretic  behavior  inherent  in  nonlinear  damping  devices.  The  DM 
model  generates  the  hysteresis  with  local  memory.  Making  use  of  the 
Duhem  operator,  the  constitutive  relation  can  be  described  by  single¬ 
valued  functions  with  two  variables  in  transformed  state  variable 
spaces.  This  makes  it  feasible  to  apply  the  force-surface  nonparametric 
identification  technique  to  hysteretic  systems.  The  Preisach  model  can 
represent  the  hysteresis  with  nonlocal  memory.  It  is  particularly 
suitable  for  describing  the  selective-memory  hysteresis  which  appears 
in  some  friction-type  isolators.  An  accurate  frequency-domain  method 
is  developed  for  analyzing  the  periodic  forced  vibration  of  hysteretic 
isolation  systems  defined  by  these  models.  A  case  study  of  wire-cable 
vibration  isolator  is  illustrated. 


1 .  Introduction 

The  dynamic  response  of  a  structure  is  highly  dependent  on 
the  ability  of  its  members  and  connections  to  dissipate  energy  by 
means  of  hysteretic  behavior.  The  assessment  of  this  behavior 
can  be  done  by  means  of  experimental  tests  and  the  use  of 
analytical  models  that  take  into  account  the  main  characteristics 
of  this  nonlinear  mechanism.  Although  a  variety  of  hysteresis 
models  have  been  proposed  in  the  past  decades,  many  structural 
systems  exhibit  more  complicated  hysteretic  performance  (mainly 
due  to  stiffness  or/ and  strength  degrading)  which  the  models  in 
existence  are  reluctant  and  even  inapplicable  to  depict  [1,2].  On 
the  other  hand,  nonlinear  vibration  isolation  has  recently  been 
recognized  as  one  of  effective  vibration  control  techniques.  In 


1043 


particular,  hysteretic  isolation  devices  have  got  wide  applications 
owing  to  their  good  diy  friction  damping  performance.  These 
hysteretic  isolators  may  exhibit  very  complicated  features  such 
as  asymmetric  hysteresis,  soft-hardening  hysteresis,  nonlocal 
selective-memory  hysteresis  [3-5].  None  of  the  models  available 
currently  in  structural  and  mechanical  areas  can  represent  all 
these  hysteresis  characteristics.  Other  more  elaborate  hysteresis 
models  need  to  be  established  for  this  purpose. 

In  reality,  hysteresis  phenomenon  occurs  in  many  different 
areas  of  science,  and  has  been  attracting  the  attention  of  many 
investigators  for  a  long  time.  However,  the  true  meaning  of 
hysteresis  varies  from  one  area  to  another  due  to  lack  of  a 
stringent  mathematical  definition  of  hysteresis.  Fortunately, 
because  of  the  applicative  interest  and  obvious  importance  of 
hysteresis  phenomenon,  Russian  mathematicians  in  1970’s  and 
the  Western  mathematicians  in  1980’s,  began  to  study  hysteresis 
systematically  as  a  new  field  of  mathematical  research  [6,7].  They 
also  deal  with  the  hysteresis  models  proposed  by  physicists  and 
engineers  in  various  areas,  but  they  separate  these  models  from 
their  physical  meanings  and  formulate  them  in  a  purely 
mathematical  form  by  introducing  the  concept  of  hysteresis 
operators.  Such  mathematical  exposition  and  treatment  can 
generalize  a  specific  model  from  a  particular  area  as  a  general 
mathematical  model  which  is  applicable  to  the  description  of 
hysteresis  in  other  areas.  In  this  paper,  two  mathematical 
hysteresis  models  are  introduced  and  the  related  problems  such 
as  identification  and  response  analysis  encountered  in  their 
application  to  nonlinear  isolation  systems  are  addressed. 

2.  Definition  of  Hysteresis 

Hysteresis  loops  give  the  most  direct  indication  of  hysteresis 
phenomena.  But  it  is  intended  here  to  introduce  a  mathematical 
definition  of  hysteresis.  Let  us  consider  a  constitutive  law:  u  r, 
which  relates  an  input  variable  u(t)  and  an  output  variable  r\t). 
For  a  structural  or  mechanical  system,  u(t)  denotes  displacement 
(strain);  r{t)  represents  restoring  force  (stress);  t  is  time.  We  can 
define  hysteresis  as  a  special  type  of  memoiy-based  relation 


1044 


between  u(t)  and  r(t).  It  appears  when  the  output  r[t)  is  not 
uniquely  determined  by  the  input  u(t)  at  the  same  instant  t,  but 
instead  r(t)  depends  on  the  evolution  of  u  in  the  interval  [0,  t]  and 
possibly  also  on  the  initial  value  ro,  i.e. 

r(t)  =  iR[u(-),ro](t)  (1) 

where  the  memory-based  functional  iR[u(-),ro](t)  is  referred  to  as 
hysteresis  operator.  In  order  to  exclude  viscosity-type  memory 
such  as  those  represented  by  time  convolution,  we  require  that  91 
is  rate-independent,  i.e.  that  r(t)  depends  just  on  the  range  of  u  in 
[0,  t]  and  on  the  order  in  which  values  have  been  attained,  not  on 
its  velocity.  In  reality,  memory  effects  may  be  not  purely  rate 
independent  as  hysteresis  is  coupled  with  viscosity- type  effects. 
However,  as  shown  later,  in  most  cases  the  rate  independent 
feature  of  hysteresis  is  consistent  with  experimental  findings, 
especially  when  evolution  (variation  in  time)  is  not  too  fast. 

3.  Duhem-Madelung  (DM)  Model 

3.1  Formulation 

The  DM  model  can  be  defined  with  or  without  referring  to  a 
confined  hysteresis  region.  For  the  structural  or  mechanical 
hysteretic  systems,  it  is  not  necessary  to  introduce  the  notion  of 
bounded  curves  because  there  exists  neither  the  saturation  state 
nor  the  major  loop.  In  this  instance,  the  DM  model  establishes  a 
mapping  (named  Duhem  operator)  91:  (u,  ro)  rby  postulating  the 


following  Cauchy  problem  [7,8] 

=  g,  (li. '■)  ■  w,  (t)  -  02 (u, r)  •  u_ (t)  (2a) 

r(0)  =  r„  (2b) 

where  an  overdot  denotes  the  derivative  with  respect  to  t;  gi{u,r] 
and  g2{ii,r)  are  referred  to  as  ascending  and  descending  functions 
(curves)  respectively;  and 

tijt)  =  max[0,u(t)]  =  t  [|u(t)|  +  u(t)]  (3a) 

u.(t)  =  min[0,u(t)]  =  ^[|u(t)|  -  u(f)]  (3b) 

Eq.  (2a)  can  be  rewritten  as 

f(f)  =  p[u,r,sgn(u)]-u(t)  (4) 


1045 


u>  0 
u<0 


(5) 


in  which  the  describing  function  has  the  form 


g{u,r,sgn{u)] 


fg{u,r,l)  =  g,{u,r) 
[g[u,r-l)  =  g^(u,r) 


It  is  obvious  that  the  DM  model  is  rate  independent.  In 
addition,  it  is  specially  noted  that  in  this  constitutive  law  the 
output  r{t)  is  not  directly  dependent  on  the  entire  history  of  u{t) 
through  [0,t];  but  instead  depends  only  on  the  local  histoiy 
covered  since  the  last  change  of  sgn(u)  and  on  the  value  of  the 
output  at  this  switching  instant.  It  means  that  the  output  can 
only  change  its  character  when  the  input  changes  direction.  As  a 
consequence,  the  DM  model  usually  represents  the  hysteresis 
with  local  memory  except  that  the  functions  gi(u,?)  and  g2{u,7)  are 
re-specified  as  hysteresis  operators. 

Within  the  framework  of  DM  formulation,  the  ascending  and 
descending  functions  gi(u,T]  and  g2(u,r)  are  just  required  to  fulfill 
suitable  regularity  conditions  and  need  not  to  be  specified  in 
specific  expressions,  so  both  the  form  and  parameters  of  the 
functions  can  be  fine-tuned  to  match  experimental  findings.  On 
the  other  hand,  the  DM  formulation  can  deduce  a  wide  kind  of 
differential-type  hysteresis  models  such  as  Bouc-Wen  model, 
Ozdemir’s  model,  Yar-Hammond  bilinear  model  and  Dahfs 


frictional  model.  For  the  Bouc-Wen  model 

r(t)  =  K.u{t)  +  z(t)  (6a) 

z{t)  =  au(t]  -  P|ti(f)|z(f)|z(t)|'”‘  -  YU(t)|z(f )|"  (6b) 

it  corresponds  to  the  DM  model  with  the  specific  ascending  and 
descending  functions  as 

grj(u,r)  =  a  +  K  “  [y  +  p  sgn(r  -Ku)]|r  -Kup  (7a) 

g2(w,r)  =  a  +  K  -  [y  -  p  sgn(r  -Kw)]|r  -ku|”  (7b) 

and  for  the  Yar-Hammond  bilinear  model 

f{t)  =  {a  - y  sgn(ii)  sgn[r  -  p  sgn(ii)]}ii  (8) 

its  describing  function  is  independent  of  u(t)  as  follows 

5f[u,r,sgn(u)l  =  gf[r,sgn(u)l  =  a  -y  sgn(u)sgn[r  -  p  sgn(u)]  (9) 


Hence,  the  Duhem  operator  also  provides  an  accessible  way 
to  construct  novel  hysteresis  models  by  prescribing  specific 


1046 


expressions  of  the  ascending  and  descending  functions. 
Following  this  approach,  it  is  possible  to  formulate  some  models 
which  allow  the  description  of  special  hysteretic  characteristics 
observed  in  experiments,  such  as  soft-hardening  hysteresis, 
hardening  hysteresis  with  overlapping  loading  envelope,  and 
asymmetric  hysteresis  [5]. 

3.2  Identification 

System  identification  techniques  are  classified  as  parametric 
and  nonparametric  procedures.  The  parametric  identification 
requires  that  the  structure  of  system  model  is  a  priori  known.  The 
advantage  of  nonparametric  identification  methods  is  that  they 
do  not  require  a  priori  the  knowledge  of  system  model.  The  most 
used  nonparametric  procedure  for  nonlinear  systems  is  the  force 
mapping  (or  called  force  surface)  method  [9].  This  method  is 
based  on  the  use  of  polynomial  approximation  of  nonlinear 
restoring  force  in  terms  of  two  variables — ^the  displacement  u(t) 
and  the  velocity  u{t) .  For  nonlinear  hysteretic  systems,  however, 
the  hysteretic  restoring  force  appears  as  a  multivalued  function 
with  respect  to  the  variables  u{t)  and  u(t)  due  to  its  history- 
dependent  and  non-holonomic  nature.  This  renders  the  force 
mapping  method  inapplicable  to  hysteretic  systems,  although 
some  efforts  have  been  made  to  reduce  the  multivaluedness  of 
the  force  surface  [10,11]. 

One  of  the  appealing  virtues  of  the  DM  model  is  that  it  can 
circumvent  this  difficulty.  Making  use  of  the  Duhem  operator, 
the  hysteretic  constitutive  relation  of  Eq.(l)  is  described  by  two 
continuous,  single-valued  functions  gi(u,r)  and  g2{u,r)  in  terms  of 
the  displacement  u(t)  and  the  restoring  force  r{t).  Thus,  single¬ 
valued  “force”  surfaces  gi{u,r)  and  g2{u,7)  can  be  formulated  in 
the  subspaces  of  the  state  variables  {u,r,gi)  and  (u,r,g2),  and  can 
be  identified  by  using  the  force  mapping  technique.  Following 
this  formulation,  a  nonparametric  identification  method  is 
developed  by  the  authors  [12].  In  this  method,  the  functions 
g\(u,7)  and  g2{u,7)  are  expressed  in  terms  of  shifted  generalized 
orthogonal  polynomials  with  respect  to  u  and  r  as  follows 

gM,r)  =  i  =  0’'(u)G<''0(r)  (10a) 

i=lj=l 


1047 


(10b) 


g^r]  =  z  i(|>,.(r)gf(i>,(u)  =  0^(u)G'^'<l>(r) 

I=lj=l 

where  =  [gf  ]^xn  =[5'zf]mxn  ^re  called  the  expansion- 

coefficient  matrices  of  gi{u,r)  and  g2{u,T].  Some  algorithms  have 
been  proposed  to  estimate  the  values  of  these  coefficient  matrices 
based  on  experimentally  observed  input  and  output  data.  It 
should  be  noted  that  here  the  vectors  0(u)  and  <^{r)  are  shifted 
generalized  orthogonal  polynomials  [13].  They  are  formulated  on 
the  basis  of  common  recurrence  relations  and  orthogonal  rule, 
and  cover  all  kinds  of  individual  orthogonal  polynomials  as  well 
as  non- orthogonal  Taylor  series.  Consequently,  they  can  obtain 
specific  polynomial-approximation  solutions  of  the  same. problem 
in  terms  of  Chebyshev,  Legendre,  Laguerre,  Jacobi,  Hermite  and 
Ultraspherical  polynomials  and  Taylor  series  as  special  cases. 

4.  Preisach  Model 


4. 1  Formulation 

The  intent  of  introducing  the  Preisach  model  is  to  supply  the 
lack  of  a  suitable  hysteresis  model  in  structural  and  mechanical 
areas,  which  is  both  capable  of  representing  nonlocal  hysteresis 
and  mathematically  tractable.  Experiments  revealed  that  the 
hysteretic  restoring  force  of  some  cable-type  vibration  isolators 
relates  mainly  to  the  peak  displacements  incurred  by  them  in  the 
past  deformation  [3].  It  will  be  shown  that  the  Preisach  model  is 
especially  effective  in  representing  such  nonlocal  but  selective- 
memory  hysteresis,  in  which  only  some  past  input  extrema  (not 
the  entire  input  variations)  leave  their  marks  upon  future  states 
of  hysteresis  nonlinearities. 

The  Preisach  model  is  constructed  as  a  superposition  of  a 
continuous  family  of  elementary  rectangular  loops,  called  relay 
hysteresis  operators  as  shown  in  Fig.  1.  That  is  [7,14], 

r(f)  = 'R[u(-)](f)  =  j||^(a,P)y„D[u(t)]dadf5  (11) 

a>P 

where  ia(a,|3)  >  0  is  a  weight  function,  usually  with  support  on  a 
bounded  set  in  the  (a,p)-plane,  named  Preisach  plane;  Ya.p[^(^)]  is 
the  relay  hysteresis  operator  with  thresholds  a  >  p.  Outputs  of 


1048 


these  simplest  hysteresis  operators  have  only  two  values  +1  and 
-1,  so  can  be  interpreted  as  two-position  relays  with  “up”  and 
“down”  positions  corresponding  to  ya,p[^^(^)l=+l  and 


1 

+1 

p 

a 

-1 

Fig.  1  Relay  Hysteresis  Operator 


Hence,  the  Preisach  model  of  Eq.(ll)  can  be  interpreted  as  a 
spectral  decomposition  of  the  complicated  hysteresis  operator  iR, 
that  usually  has  nonlocal  memory,  into  the  simplest  hysteresis 
operators  7ct,p  with  local  memory.  In  the  following,  we  illustrate 
how  the  model  depicts  the  nonlocal  selective-memory  feature. 

Consider  a  triangle  T  in  the  half-plane  a  >  (3  as  shown  in 
Fig.2.  It  is  assumed  that  the  weight  function  )i(a,p)  is  confined  in 
the  triangle  T,  i.e.  }i(a,p)  is  equal  to  zero  outside  T.  Following  the 
Preisach  formulation,  at  any  time  instant  t,  the  triangle  T  can  be 
subdivided  into  two  sets:  S^(t)  consisting  of  points  (a,P)  for  which 
the  corresponding  Yc^.p-operators  are  in  the  “up”  position;  and  S^(t) 
consisting  of  points  {a,P)  for  which  the  corresponding  Ya.p“ 
operators  are  in  the  “down”  position.  The  interface  L(t)  between 
S^(t)  and  Sr(t)  is  a  staircase  line  whose  vertices  have  a  and  p 
coordinates  coinciding  respectively  with  local  maxima  and 
minima  of  input  at  previous  instants  of  time.  The  nonlocal 
selective-memory  is  stored  in  this  way.  Thus,  the  output  r(^)  a-t 
any  instant  t  can  be  expressed  equivalently  as  [14] 

r{t)=  J|^(a,P)dadp-  |jM,(a,P)dadp  (12) 

S'(£)  S'(t) 

It  should  be  noted  that  the  Preisach  model  does  not 
accumulate  all  past  extremum  values  of  input.  Some  of  them  can 
be  wiped  out  by  sequent  input  variations  following  the  wiping-out 
property  {deletion  rule):  each  local  input  maximum  wipes  out  the 


1049 


vertices  of  L(t)  whose  a-coordinates  are  below  this  maximum,  and 
each  local  minimum  wipes  out  the  vertices  whose  p-coordinates 
are  above  this  minimum.  In  other  words,  only  the  alternating 
series  of  dominant  input  extrema  are  stored  by  the  Preisach 
model;  all  other  input  extrema  are  erased. 


Fig.  2  Input  Sequence  and  Preisach  Plane 


4.2  Identification 


It  is  seen  from  Eq.(l  1)  that  the  Preisach  model  is  governed  by 
the  weight  function  |i(a,p)  after  determining  L(t)  which  depends 
on  the  input  sequence.  !i(a,P)  is  a  single-valued  function  with 
respect  to  two  variables  a  and  p.  Hence,  the  aforementioned 
nonparametric  identification  method  can  be  also  implemented  to 
identify  iLL(a,p)  by  expanding  it  in  a  similar  expression  to  Eq.(lO). 
An  alternating  approach  is  to  define  the  following  function 


H'(a',P')  =  |j|a(a,p)dadp  =  ^.[^  ^(ot,P)da]dp  (13) 


r(cc'.3') 


where  T(a',p')  is  the  triangle  formed  by  the  intersection  of  the 
line  a  =  a' ,  p  =  p'  and  a  =  p .  Differentiating  Eq.(13)  yields 


^(a',P')  =  -- 


aa'ap' 


(14) 


Thus,  the  force  mapping  identification  technique  can  be  applied 
to  determine  H(a,p)  consistent  with  the  experimental  data,  and 
then  ^(a,p)  is  obtained  by  Eq.(14). 


5.  Steady-State  Response  Analysis 

Hysteretic  systems  are  strongly  nonlinear.  A  study  of  the 
steady- state  oscillation  is  one  of  the  classical  problems  of 


1050 


nonlinear  systems.  Usually,  the  dynamic  behavior  of  a  nonlinear 
system  is  represented  by  its  resonant  frequency  and  frequency 
response  characteristics.  In  the  following,  an  accurate  frequency- 
domain  method  accommodating  multiple  harmonics  is  developed 
to  analyze  the  periodically  forced  response  of  hysteretic  systems 
defined  by  mathematical  hysteresis  models. 

Fig. 3  shows  a  single-degree-of- freedom  hysteretic  oscillator 
with  mass  m,  viscous  damping  coefficient  c,  and  linear  stiffness 
ky  subjected  to  an  external  excitation  F(t),  for  which  the  governing 
equation  of  motion  is 

m  •  u(t)  +  c  •  u(t)  +  k  '  u[t)  +  r{t)  =  F(t)  (15) 

where  the  hysteretic  restoring  force  r(t)  is  represented  by  the  DM 
model  as  Eq.(4).  It  is  worth  noting  that  for  the  kinetic  equation 
Eq.(15),  the  excitation  is  F(t)  and  the  response  is  u(t);  and  for  the 
hysteretic  constitutive  law  Eq.(4),  u(t)  is  input  and  r{t)  is  output. 
The  causal  relationship  is  different. 


Fig.  3  Single -Degree-of-Freedom  Hysteretic  System 

Due  to  the  hereditary  nature  of  the  hysteresis  model,  it  is 
difficult  to  directly  solve  the  kinetic  equation  Eq.(15)  by  iteration. 
Here,  Eq.(15)  is  only  used  to  establish  the  relation  between  the 
harmonic  components  of  u(t)  and  r(t).  Suppose  that  the  system  is 
subjected  to  a  general  periodic  excitation  F[t)  with  known 
harmonic  components  F={Fq  F^  F2  •••  F2  .  The  multi¬ 
harmonic  steady-state  response  can  be  expressed  as 

a  ^  ^  . 

u{t)  =  ~ +  a j  cos  j(Dt +  a j  sin  jcot  (16) 

2  j=i 


1051 


in  which  a={ao  a2  •••  a^v  ^2  unknown 

vector  containing  the  harmonic  components  of  u(t).  Introducing 

Eq.(16)  into  Eq.(15)  and  using  the  Galerkin  method  provide 

ro^F^-k-a^  (17a) 

Tj  =  Fj  -c-citj-a]  [j  =  1,2,  •••  ,  N)  (17b) 

Tj  =Fj  +c-coj-a^.  -(/c-m-coV^)-a*  (j  =  1,  2,  ,N)  (17c) 

where  r={ro  q  •••  r^}'^  is  the  harmonic  vector  of 

the  hysteretic  restoring  force  r(t).  Referring  to  the  hysteretic 
constitutive  law,  we  define  the  determining  equation  as 

D(t)  =  r{t)  -  g[u,r,sgn(u)]  •  u{t)  (18) 

When  a  is  the  solution  of  u(t),  applying  the  Galerkin  method 
into  Eq.(18)  and  considering  Eq.(17)  achieve 

d{a)  =  0  (19) 

where  the  vector  d{a)={dQ  ^3  --dj^  d^  d^  "•  is  comprised 

of  the  harmonic  components  of  D(t)  corresponding  to  a.  An 

efficient  procedure  to  seek  the  solution  of  Eq.(19)  is  the 
Levenberg-Marquardt  algorithm  with  the  iteration  formula 

where  the  Jacobian  matrix  J[a(^)]  =  dd(a) / da\a=a^^)  ;  9^  is  the 

Levenberg-Marquardt  parameter  and  I  is  identity  matrix. 

At  each  iteration,  the  function  vector  and  Jacobian 

matrix  should  be  recalculated  with  updated  values  of 

Here,  a  frequency/ time  domain  alternation  scheme  by  FFT  is 
introduced  to  evaluate  the  values  of  d(a)  and  J[a)  at  d{a) 

and  dd[a)/da  are  known  to  be  the  Fourier  expansion  coefficients 
of  D(t)  and  dD(t)/ da  respectively.  For  a  given  a(^)  and  known  F,  the 
corresponding  r[a(^)]  is  obtained  from  Eq.(17),  and  the  inverse 
FFT  is  implemented  for  and  r[a(^)]  to  obtain  all  the  time 
domain  discrete  values  of  u{t] ,  u(t) ,  r(t)  and  f(t)  over  an  integral 
period.  Then  the  time  domain  discrete  values  of  the  function 
D{u,u,r,r,t) ,  corresponding  to  a=a(^),  are  evaluated  from  Eq.(18). 
Making  use  of  forward  FFT  to  these  time  domain  discrete  values 
of  D{u,u,r,r,t) ,  the  values  of  function  vector  d[a(^)]  are  obtained. 


1052 


Similarly,  the  partial  differential  dD{t)  /  da  can  be  analytically 
evaluated  in  the  time  domain.  Forward  FFT  to  the  time  domain 
values  of  dD(t)  /  da  at  ct=a(^)  gives  rise  to  dd[ai^)]/da. 

6.  Case  Study 

Wire-cable  vibration  isolators  are  typical  hysteretic  damping 
devices.  Dynamic  tests  show  that  their  hysteresis  behaviors  are 
almost  independent  of  the  frequency  in  the  tested  frequency 
range  [4,15].  Experimental  study  and  parametric  modelling  of  a 
wire-cable  isolator  have  been  carried  out  [5].  Fig. 4  shows  the 
experimental  hysteresis  loops  in  shear  mode.  It  is  seen  that  for 
relatively  small  deformations,  the  isolator  exhibits  softening 
hysteresis  loops.  When  large  displacements  are  imposed,  the 
stiffness  of  the  loops  becomes  smoothly  hard.  This  nature  is 
referred  to  as  soft-hardening  hysteresis.  Based  on  the  Bouc-Wen 
model,  a  parametric  identification  was  performed  to  model  these 
hysteresis  loops,  but  the  result  is  unsatisfactory.  This  is  due  to 
the  fact  that  the  Bouc-Wen  model  cannot  represent  such  soft- 
hardening  nature  of  hysteresis. 


Fig.  4  Experimental  Hysteresis  Loops 


We  now  use  the  DM  model  to  represent  these  hysteresis 
loops,  and  perform  a  nonparametric  identification  to  determine 
the  functions  gi(u,7)  and  g2(u,7).  The  simplest  Taylor  series  are 
adopted,  i.e.  (i),(r)  =  (r  /  and  (|)j(u)  =  (u  j  UqY~^  (ro  =  20.0  and 
uo  =  2.0).  Fig. 5  shows  the  identified  “force”  surfaces  of  gi{u,T)  and 
g2(u,r)  by  taking  m=n~S.  Fig. 6  presents  the  theoretical  hysteresis 


1053 


loops  generated  by  the  DM  model  using  the  identified  g\(u,T)  and 
g2(u,}).  It  is  seen  that  the  modeled  hysteresis  loops  are  agreeable 
to  the  observed  loops.  In  particular,  the  soft-hardening  nature  is 
reflected  in  the  modeled  hysteresis  loops. 


-200 


Fig.  6  Modeled  Hysteresis  Loops  by  DM  Model 


After  performing  the  modelling  of  hysteretic  behavior,  the 
dynamic  responses  of  hysteretic  systems  can  be  predicted  by  the 
developed  method.  Fig. 7  shows  a  vibration  isolation  system 
installed  with  wire-cable  isolators  in  shear  mode.  It  is  subjected 
to  harmonic  ground  acceleration  excitation  Xg[t)=Acos2Kft.  The 

equation  of  motion  of  the  system  is  expressed  as 

m-u(t)  + K  ■r(t)  = -rri' Xg(t)  (21) 

where  m  is  the  mass  of  the  system;  K  is  number  of  the  isolators 
installed.  u{t)  is  the  displacement  of  the  system  relative  to  the 
ground.  r(t)  is  the  restoring  force  of  each  isolator  and  has  been 
determined  from  nonparametric  identification. 

Fig. 8  illustrates  the  predicted  frequency-response  curves  of 
the  relative  displacement  when  7n=6kg  and  K=2.  The  excitation 
amplitude  A  is  taken  as  0.25g,  O.SOg,  0.35g,  0.40g  and  0.45g 
respectively.  The  frequency-response  curves  show  clearly  the 
nonlinear  nature  of  the  wire-cable  isolation  system. 


i 

I 


Xg(t) 


Fig.  7  Vibration  Isolation  System  with  Wire -Cable  Isolators 


Fig.  8  Frequency  Response  Curves  of  Relative  Displacement 


1055 


7.  Concluding  Remarks 


This  paper  reports  a  preliminaiy  work  of  introducing  the 
mathematical  hysteresis  models  in  structural  and  mechanical 
areas.  It  is  shown  that  a  wide  kind  of  differential  hysteresis 
models,  which  are  extensively  used  at  present,  can  be  derived 
from  the  Duhem-Madelung  (DM)  model.  Thus,  the  mathematical 
properties  concerning  the  DM  model  are  also  possessed  by  these 
models.  Two  potential  advantages  appear  when  the  DM 
formulation  is  used.  Firstly,  it  allows  to  apply  the  force  mapping 
technique  to  hysteretic  systems.  Secondly,  it  provides  an 
approach  to  construct  novel  differential  models  which  reflect 
some  special  hysteretic  characteristics.  The  Preisach  model  is 
shown  to  be  capable  of  representing  nonlocal  hysteresis  and 
mathematically  tractable.  It  offers  a  more  accurate  description  of 
several  observed  hysteretic  phenomena.  Emphasis  is  placed  on 
demonstrating  the  selective-memory  nature  of  this  hysteresis 
model.  The  case  study  based  on  experimental  data  of  a  wire- 
cable  isolator  has  shown  the  applicability  of  the  mathematical 
hysteresis  model,  and  the  validity  of  the  steady-state  response 
analysis  method  proposed  in  the  present  paper. 

Acknowledgment:  This  study  was  supported  in  part  by  the 
Hong  Kong  Research  Grants  Council  (RGC)  and  partly  by  The 
Hong  Kong  Polytechnic  University.  These  supports  are  gratefully 
acknowledged. 


References 

1.  Azevedo,  J.  and  Calado,  L.,  ‘‘Hysteretic  behaviour  of  steel 
members:  analytical  models  and  experimental  tests”,  J. 
Construct  Steel  Research,  1994,  29,  71-94. 

2.  Kayvani,  K.  and  Barzegar,  F.,  “Hysteretic  modelling  of  tubular 
members  and  offshore  platforms”,  Eng.  Struct,  1996,  18,  93- 
101. 

3.  Lo,  H.R.,  Hammond,  J.K.  and  Sainsbury,  M.G.,  “Nonlinear 
system  identification  and  modelling  with  application  to  an 
isolator  with  hysteresis”,  Proc.  6th  Int  modal  Anal  Conf., 
Kissimmee,  Florida,  1988,  Vol.II,  1453-1459. 


1056 


4.  Demetriades,  G.F.,  Constantinou,  M.C.  and  Reinhorn,  A.M., 
“Study  of  wire  rope  systems  for  seismic  protection  of 
equipment  in  buildings”,  Eng.  Struct,  1993,  15,  321-334. 

5.  Ni,  Y.Q.,  “Dynamic  response  and  system  identification  of 
nonlinear  hysteretic  systems”,  PhD  Dissertation,  The  Hong 
Kong  Polytechnic  University,  Hong,  Kong,  November  1996. 

6.  Krasnosefskii,  M.A.  and  Pokrovskii,  A.V.,  Systems  with 
Hysteresis,  translated  from  the  Russian  by  M.  Niezgodka, 
Springer-Verlag,  Berlin,  1989. 

7.  Visintin,  A.,  Differential  Models  of  Hysteresis,  Springer-Verlag, 
Berlin,  1994, 

8.  Macki,  J.W.,  Nistri,  P.  and  Zecca,  P.,  “Mathematical  models 
for  hysteresis”,  SIAM  Review,  1993,  35,  94-123. 

9.  Masri,  S.F.  and  Caughey,  T.K.,  “A  nonparametric 

identification  technique  for  nonlinear  dynamic  problems”,  J. 
Appl  Mech,  ASME,  1979,  46,  433-447. 

10.  Lo,  H.R.  and  Hammond,  J.K.,  “Nonlinear  system  identification 
using  the  surface  of  nonlinearity  form:  discussion  on 
parameter  estimation  and  some  related  problems”,  Proc.  3rd 
Int  Conf  Recent  Adv.  Struct.  Dyn.,  Southampton,  UK,  1988, 
339-348. 

11. Benedettini,  F.,  Capecchi,  D.  and  Vestroni,  F.,  “Identification 
of  hysteretic  oscillators  under  earthquake  loading  by 
nonparametric  models”,  J.  Eng.  Mech.,  ASCE,  1995,  121,  606- 
612. 

12.  Ni,  Y.Q.,  Ko,  J.M.  and  Wong,  C.W.,  “Modelling  and 
identification  of  nonlinear  hysteretic  vibration  isolators”. 
Accepted  to  SPJE’s  4th  Annual  Symposium  on  Smart  Structures 
and  Materials  :  Passive  Damping  and  Isolation,  3-6  March 
1997,  San  Diego,  USA. 

13. Ni,  Y.Q.,  Wong,  C.W.  and  Ko,  J.M.,  “The  generalized 
orthogonal  polynomial  (GOP)  method  for  the  stability  analysis 
of  periodic  systems”,  Proc.  Int.  Conf.  Comput.  Methods  Struct. 
Geotech.  Eng.,  Hong  Kong,  1994,  Vol.II,  464-469. 

14. Mayergoyz,  I.D.,  Mathematical  Models  of  Hysteresis,  Springer- 
Verlag,  New  York,  1991. 

15.  Ko,  J.M.,  Ni,  Y.Q.  and  Tian,  Q.L.,  “Hysteretic  behavior  and 
empirical  modeling  of  a  wire-cable  vibration  isolator”,  Int.  J. 
Anal.  Exp.  Modal  Anal,  1992,  7,  111-127. 


1057 


1058 


The  identification  of  turbogenerator  foundation  models 
from  run-down  data 


M  Smart,  M  I  Friswell,  A  W  Lees,  U  Prells 

Department  of  Mechanical  Engineering 
University  of  Wales  Swansea,  Swansea  SA2  8PP  UK 
email:  m.smart@swansea.ac.nk 


ABSTRACT 

The  trend  of  placing  turbines  in  modern  power  stations  on  flexi¬ 
ble  steel  foundations  means  that  the  foundations  exert  a  considerable 
influence  on  the  dynamics  of  the  system.  In  general,  the  complex¬ 
ity  of  the  foundations  means  that  models  are  not  available  a  priori, 
but  rather  need  to  be  identified.  One  way  of  doing  this  is  to  use 
the  measured  responses  of  the  foundation  at  the  bearings  to  the 
synchronous  excitation  obtained  when  the  rotor  is  run  down.  This 
paper  discusses  the  implementation  of  such  an  estimation  technique, 
based  on  an  accurate  model  of  the  rotor  and  state  of  unbalance,  and 
some  knowledge  of  the  dynamics  of  the  bearings.  The  effect  of  errors 
in  the  bearing  model  and  response  measurements  on  the  identified 
parameters  is  considered,  and  the  instrumental  variable  method  is 
suggested  as  one  means  of  correcting  them. 


1  INTRODUCTION 

The  cost  of  failure  of  a  typical  turbine  in  a  modern  power  station  is 
very  high,  and  therefore  development  of  condition  monitoring  techniques 
for  such  machines  is  an  active  area  of  research.  Condition  monitoring  relies 
on  measuring  machine  vibrations  and  using  them  to  locate  and  quantify 
faults,  which  obviously  requires  an  accurate  dynamic  model  of  the  ma¬ 
chine.  Although  the  dynamic  characteristics  of  rotors  are  generally  well 
understood,  the  foundations  on  which  they  rest  are  not.  Since  the  founda¬ 
tions  are  often  quite  flexible,  they  can  contribute  considerably  to  the  rotor’s 
dynamic  behaviour. 

Finite  element  modelling  has  been  attempted  but  the  complexity  of 
the  foundations,  and  the  fact  that  they  often  differ  substantially  from  the 
original  drawings  rendered  the  technique  generally  unsuccessful [1].  Exper¬ 
imental  modal  analysis  is  another  possible  solution,  but  this  requires  that 
the  rotor  be  removed  from  the  foundation,  and  that  all  casings  remain  in 
place,  which  is  not  practical  for  existing  power  plant.  However,  mainte¬ 
nance  procedures  require  that  rotors  are  run  down  at  regular  intervals  and 
this  procedure  provides  forcing  to  the  foundation  over  the  frequency  range 


1059 


of  operation.  By  measuring  the  response  at  the  bearing  pedestals  (which 
is  already  performed  for  condition  monitoring  purposes)  an  input-output 
relation  for  the  foundation  may  be  obtained. 

Lees  [2]  developed  a  least-squares  method  to  calculate  the  foundation 
parameters  by  assuming  that  an  accurate  model  exists  for  the  rotor,  that 
the  state  of  unbalance  is  known  from  balancing  runs,  and  that  the  dynamic 
stiffness  matrices  of  the  bearings  can  be  calculated.  Although  bearing  mod¬ 
els  are  not  in  fact  well  characterised,  Lees  and  Friswell[3]  showed  that  the 
parameter  estimates  are  only  sensitive  to  the  bearing  stiffnesses  over  limited 
frequency  ranges,  which  can  be  calculated. 

Feng  and  Hahn [4]  followed  a  similar  approach  but  added  extra  informa¬ 
tion  by  measuring  the  displacements  of  the  shaft.  Zanetta[5]  also  measured 
the  shaft  displacements  but  included  the  bearing  characteristics  as  param¬ 
eters  to  be  estimated.  Although  any  extra  information  is  desirable  in  a 
parameter  estimation  routine  the  equipment  necessary  to  measure  these 
quantities  only  exists  in  the  newer  power  stations,  and  it  was  desired  to 
make  the  method  applicable  to  older  plant  as  well.  In  the  analysis  presented 
here,  the  measured  data  consists  of  the  motion  of  the  bearing  pedestals  in 
the  horizontal  and  vertical  directions,  although  not  necessarily  in  both  di¬ 
rections  at  every  bearing. 

2  THEORY 

2.1  Force  estimation 

If  D  is  the  dynamic  stiffness  matrix  of  a  structure  defined  as 

D{u)  K  ^luC  -oj^M  (1) 

where  M,C,K  are  the  mass,  damping  and  stiffness  matrices  then 

Dx  =  f  (2) 

where  x  is  response  and  f  force.  Referring  to  figure  1  it  is  seen  that  the  rotor 
is  connected  to  the  foundation  via  the  bearings.  It  is  assumed  that  good 
models  exist  for  both  rotor  and  bearings,  and  that  the  state  of  unbalance 
is  known.  The  implications  of  these  assumptions  will  be  discussed  later. 

The  dynamic  stiffness  equation  for  the  whole  system  may  be  written  as 


The  subscripts  r  and  /  refer  to  the  rotor  and  foundation  degrees-of-freedom 
respectively,  u  refers  to  the  unbalance  forces  and  b  to  the  bearing  forces. 


1060 


ROTOR 


Figure  1:  Rotor-bearing  system 


There  is  a  negative  sign  before  the  bearing  forces  ft,  since  they  refer  to  the 
forces  acting  on  the  bearings.  The  foundation  d.o.f  are  those  where  the  re¬ 
sponses  are  measured,  in  other  words  no  internal  d.o.f  are  represented.  Df 
therefore  represents  a  reduced  dynamic  stiffness  matrix.  The  response  mea¬ 
surements  will  not  be  the  total  vibration  level  at  the  bearings  but  rather  the 
vibrations  at  once-per-revolution  and  it  is  assumed  that  no  dynamic  forces 
at  this  frequency  will  be  transmitted  to  the  foundation  via  the  substructure 
onto  which  it  is  fixed. 

The  equation  for  the  bearings  in  the  global  coordinate  system  is 


/  Dbrr  Dbrf  W  ^7-  ^  ^  fbT\ 
V  Dij,  Dill  )\^s)  \fbs) 


(4) 


This  assumes  that  the  bearings  behave  as  complex  springs,  in  other 
words  they  have  negligible  inertia  and  no  internal  d.o.f.  Substituting  (4) 
into  (3)  we  have 


/  Dt„  Dtrl  \  (  ^'■  \  =  ( 

V  A/r  A//  J  [  J  A  °  / 

where 

/  Dtrl  0  \  f  Dirr  Di^l  \\ 

[Dtlr  DtllJ-[[  0  DlJ  +  [Diir  Dill)) 


(5) 

(6) 


1061 


and  where  subscript  t  refers  to  the  total  model.  Solving  equation  (5)  for 
Xr  leads  to 

Xr  —  J^trr  ifur  ~~ 

and  solving  equation  (4)  for  ft /  yields 

fbf  —  ^fhr  ~  ~^brr^r  ~  ^brf^f  (S) 

All  quantities  in  equation  (8)  are  known  either  from  assumed  models  {Dr,  D^) 
or  experiment  {xf,  /^r)-  This  calculated  force  fbf  may  then  be  used  to¬ 
gether  with  the  measured  responses  to  estimate  the  foundation  parameters. 

2.2  Foundation  parameter  estimation 

Once  the  forces  have  been  estimated,  the  foundation  parameters  must 
be  derived.  The  dynamic  stiffness  equation  for  the  foundation  is 


DfXf  —  fbf  (9) 

Although  D/  is  a  reduced  stiffness  matrix  it  is  assumed  that  it  has  the 
form  of  equation  (1).  Therefore  equation  (9)  may  be  written  as 

W{u)v  =  fbf  (10) 

where  is  a  column  vector  formed  from  the  elements  of  M,  C  and  K  and 
W  is  a  matrix  formed  from  the  response  vector  which  depends  explicitly  on 
u.  Clearly  this  is  an  under-determined  set  of  equations,  but  by  taking  mea¬ 
surements  at  many  frequencies  it  may  be  made  over-determined,  and  thus 
solvable  in  a  least  squares  sense.  Since  the  magnitude  of  the  mass,  damp¬ 
ing  and  stiffness  elements  normally  differ  by  several  orders-of-magnitude,  it 
was  found  expedient  to  scale  the  mass  parameters  by  o;^,  and  the  damping 
parameters  by  uJ,  where  u  is  the  mean  value  of  the  frequency. 

2.3  Errors  in  estimates 

It  is  necessary  now  to  examine  the  effect  of  errors  on  the  parameter  esti¬ 
mates.  Equation  (10)  is  of  the  form  Ax  =  b,  where  A  has  dimension  mxn. 
In  this  particular  case,  A  depends  on  the  measured  response  Xf,  whilst  b 
depends  on  the  measured  response,  applied  unbalance,  and  assumed  rotor 
and  bearing  models.  Therefore  the  estimated  parameters  will  be  sensitive 
to  the  following  errors: 

1.  Errors  in  the  rotor  model 

2.  Errors  in  the  bearing  model 

3.  Errors  in  the  state  of  unbalance 


1062 


4.  Errors  in  the  measured  foundation  response 

The  rotor  model  is  generally  well  known,  as  is  the  state  of  unbalance, 
so  the  main  source  of  error  in  the  estimates  is  due  to  measurement  noise 
and  bearing  uncertainty.  If  the  least  squares  problem  is  formulated  as 


Ax  =  b  (11) 

then 

(Aq  +  Ajv)ic  =  bo  +  b/i/  (12) 

where  the  subscript  N  refers  to  noise  and  0  to  data  which  is  noise-free.  The 
least-squares  estimate  is  given  by 

X  =  (Aq  Ao  +  A^Aiv  +  +  A^Ao)  ^(Aq  +  Ayv)^(bo  +  b;v)  (13) 

Even  if  the  noise  on  the  outputs  is  uncorrelated  with  the  noise  on  the  inputs 
the  expected  value  of  x  does  not  equal  that  of  its  estimate: 

E[x]  -  E[x]  0  (14) 

In  other  words  the  estimate  x  is  biased  [6].  In  order  to  reduce  the  bias  of 
the  estimates,  the  instrumental  variables  method  can  be  used.  Essentially, 
it  requires  the  use  of  a  matrix  that  is  uncorrelated  with  the  noise  on  the 
outputs,  but  which  is  strongly  correlated  with  the  noise-free  measurements 
themselves.  If  W  is  the  instrumental  variable  matrix,  then 

W'^Ax  =  W'^b  (15) 

Expanding 

i  =  (W^Ao  +  +  b^)  (16) 

This  means  that  E[x]  =  x,  in  other  words  unbiased  estimates  result. 
Fritzen[7]  suggested  an  iterative  method  for  solving  for  the  parameters. 
Initially,  equation  (10)  is  solved  in  a  least-squares  sense,  and  the  values 
of  the  estimated  parameters  are  used  to  calculate  outputs  for  the  model. 
These  outputs  are  then  used  to  create  W  in  the  same  way  as  the  original 
outputs  were  used  to  create  A,  new  estimates  are  obtained,  and  if  neces¬ 
sary  the  process  is  repeated.  Experience  seems  to  suggest  good  convergence 
properties[7]. 


3  SIMULATION 

The  method  under  discussion  in  this  paper  was  tested  on  a  model  of  a 
small  test  rig  located  at  Aston  University,  Birmingham.  This  consists  of 
a  steel  shaft  approximately  1.1m  long,  with  nominal  diameter  38mm.  The 
shaft  is  supported  at  either  end  by  a  journal  bearing  of  diameter  100mm, 


1063 


1  2  3  4  5  6  7  8  9  10  11  12 


Figure  2:  Rotor-bearing  system 

L/D  ratio  of  0.3  and  clearance  of  25/.tm.  There  are  two  shrink-fitted  bal¬ 
ancing  discs  for  balancing  runs.  Each  bearing  is  supported  on  a  flexible 
pedestal  to  simulate  the  flexible  foundations  encountered  in  power  station 
turbines.  At  present  these  pedestals  are  bolted  onto  a  massive  lathe  bed. 
The  rotor  is  powered  by  a  DC  motor  attached  via  a  belt  to  a  driving  pulley, 
which  is  in  turn  attached  via  a  flexible  coupling  to  the  main  rotor  shaft. 

A  schematic  of  the  rig  is  shown  in  figure  2.  Dimensions  of  each  station 
and  material  properties  are  given  in  table  1.  A  finite  element  model  of  the 
rotor  with  23  elements  was  created  and  short  bearing  theory  was  used  to 
obtain  values  for  the  bearing  stiffness  and  damping[8]. 

The  pedestals  themselves  consist  of  two  rectangular  steel  plates,  600mm 
X  150mm  which  have  two  channels  cut  into  them,  and  which  are  supported 
on  knife-edges  (figure  3).  The  vertical  stiffness  arises  from  the  hinge  effect 
of  the  channels,  whilst  the  horizontal  stiffness  is  as  a  result  of  the  shaft 
centre  tilting  under  an  applied  load.  Treating  the  supports  as  beams,  the 
theoretical  stiffnesses  are: 

Ky  =  0.^5MN/m  !<:,  =  1.5MiV/m 

where  x  and  y  refer  to  the  horizontal  and  vertical  directions  respectively. 
The  masses  and  damping  factors  were  taken  as: 

il/4  =  My  =  50/cp  Ca:  =  Cy  =  150iV  •  s/m 

The  estimation  theory  was  tested  using  this  model.  The  finite  element 
model  was  used  to  generate  responses  at  the  bearings  for  frequencies  from 


1064 


Table  1:  Table  of  rotor  rig  properties 


Shaft  Properties 

Station 

Length  (mm) 

Diameter  (mm) 

E  (GPa) 

P  (kg/m^) 

1 

6.35 

38.1 

200 

7850 

2 

25.4 

77.57 

200 

7850 

3 

50.8 

38.1 

200 

7850 

4 

203.2 

100 

200 

7850 

5 

177.8 

38.1 

200 

7850 

6 

50.8 

116.8 

200 

7850 

7 

76.2 

38.1 

200 

7850 

8 

76.2 

109.7 

200 

7850 

9 

76.2 

38.1 

200 

7850 

10 

50.8 

102.9 

200 

7850 

11 

177.8 

38.1 

200 

7850 

12 

203.2 

100 

200 

7850 

Balancing  discs 

Station 

Length  (mm) 

Diameter  (mm)  Unbalance  (kg  •  m) 

6 

25.4 

203.2 

0.001 

25.4 

203.2 

0.001 

0  to  30  Hz  with  a  spacing  of  0.1  Hz.  The  responses  were  corrupted  by 
normally  distributed  random  noise  with  zero  mean  and  standard  deviation 
of  0.1%  of  the  maximum  response  amplitude  (applied  to  both  real  and 
imaginary  parts  of  the  response).  At  each  frequency  the  bearing  static 
forces  were  disturbed  by  noise  drawn  from  a  uniform  distribution  spanning 
an  interval  of  20%  of  the  force  magnitude,  to  introduce  uncertainty  into  the 
bearing  parameters.  The  unbalance  was  assumed  to  be  exactly  known.  A 
series  of  30  runs  was  performed,  foundation  parameter  estimates  calculated 
and  the  mean  and  standard  deviations  of  these  estimates  obtained. 

The  magnitudes  of  the  responses  at  both  bearings  are  given  in  figure  4, 
which  show  that  there  are  four  critical  speeds  in  the  frequency  range  under 
consideration.  A  sample  of  true  and  estimated  forces  in  the  bearings  are 
shown  in  figures  5  and  6. 

The  means  {fi)  and  standard  deviations  (a)  of  the  least-square  (LS) 
and  instrumental  variable  (IV)  estimates  for  the  foundation  parameters 
are  shown  in  table  2. 


Displacement  (m) 


Figure  3:  Flexible  bearing  supports 


Figure  4:  Magnitudes  of  responses  at  bearings 


1066 


Table  2;  Parameter  estimates  for  foundations  showing  uncertainty 


1069 


4  DISCUSSION 


The  results  in  table  2  show  a  clear  improvement  in  parameter  estimates 
when  the  instrumental  variable  method  is  used.  There  is  a  clear  bias  in 
the  least-squares  estimates  which  is  significantly  less  when  the  instrumen¬ 
tal  variable  method  is  employed.  Also,  despite  the  fact  that  the  bearing 
parameters  are  assumed  to  be  seriously  in  error,  the  estimates  appear  to  be 
insensitive  to  them.  This  will  be  true  provided  that  the  bearings  are  much 
stiffer  than  the  foundation  (a  reasonable  assumption  in  practice).  It  does 
appear  however  that  in  some  cases  the  standard  deviation  of  the  instru¬ 
mental  variable  estimate  is  larger  than  that  of  the  least-squares  estimate, 
a  fact  which  warrants  further  investigation. 

As  far  as  the  rotor  model  is  concerned,  impact  tests,  which  are  per¬ 
formed  on  rotors  prior  to  them  entering  service,  normally  give  experimental 
frequencies  which  are  within  a  few  percent  of  the  theoretical  ones.  Thus 
the  assumption  that  the  rotor  model  is  accurately  known  would  appear  to 
be  reasonable. 

The  state  of  unbalance  may  in  theory  be  established  from  a  balancing 
run.  If  two  successive  run-downs  are  performed,  one  due  to  the  unknown 
system  unbalance  and  one  with  known  balance  weights  attached,  then  pro¬ 
vided  the  system  is  linear  the  response  measurements  may  be  vectorially 
subtracted  to  give  the  response  due  to  the  known  balance  weights  alone. 
In  order  to  ascertain  the  effect  of  unbalance  uncertainty  on  the  parameter 
estimates,  one  run  was  performed  assuming  no  error  in  the  unbalance.  It 
should  be  noted  that  this  assumes  that  the  system  is  linear. 

5  CONCLUSIONS 

A  method  of  estimating  turbogenerator  foundation  parameters  from 
potentially  noisy  measurement  data  is  demonstrated.  It  is  shown  that 
making  use  of  the  instrumental  variable  method  reduces  the  bias  in  the 
estimates  and  improves  them  quite  significantly. 

6  ACKNOWLEDGEMENT 

The  authors  wish  to  acknowledge  the  support  and  funding  of  Nuclear 
Electric  Ltd  and  Magnox  Electric  Pic.  Dr  Friswell  wishes  to  acknowledge 
the  support  of  the  Engineering  and  Physical  Sciences  Research  Council 
through  the  award  of  an  advanced  fellowship. 


1070 


REFERENCES 


[1]  A.  W.  Lees  and  I.  C.  Simpson.  The  dynamics  of  turbo-alternator  foun¬ 
dations:  Paper  C6/83.  In  Conference  on  steam  and  gas  turbine  founda¬ 
tions  and  shaft  alignment,  Bury  St  Edmunds,  1983,  IMechE,  pp37-44. 

[2]  A.  W.  Lees.  The  least  squares  method  applied  to  identify  ro¬ 
tor/foundation  parameters:  Paper  C306/88.  In  Proceedings  of  the  Inter¬ 
national  Conference  on  Vibrations  in  Rotating  Machinery,  Edinburgh, 
1988,  IMechE,  pp209-216. 

[3]  M.  1.  Friswell  and  A.  W.  Lees.  Estimation  of  forces  exerted  on  machine 
foundations.  In  M.  1.  Friswell  and  J.  E.  Mottershead,  editors,  Interna¬ 
tional  Conference  on  Identification  in  Engineering  Systems,  Swansea, 
1996,  pp793-803. 

[4]  N.  S.  Feng  and  E.  J.  Hahn.  Including  foundation  effects  on  the  vibra¬ 
tion  behaviour  of  rotating  machinery.  Mechanical  Systems  and  Signal 
Processing,  1995,  9,  pp243-256. 

[5]  G.  A.  Zanetta.  Identification  methods  in  the  dynamics  of  turbogener¬ 
ator  rotors:  Paper  C432/092.  In  IMechE  Conference  on  Vibrations  in 
rotating  machinery,  Bath,  1992.  IMechE,  ppl73-181. 

[6]  J.  Schoukens  and  R.  Pintelon.  Identification  of  linear  systems.  Perga- 
mon  Press,  1991. 

[7]  C.  P.  Fritzen.  Identification  of  mass,  damping  and  stiffness  matrices  of 
mechanical  systems.  Journal  of  Vibration,  Acoustics,  Stress  and  Relia¬ 
bility  if  Design,  1986  108,  pp9-17. 

[8]  D.  M.  Smith.  Journal  bearings  in  Turbomachinery.  Chapman  and  Hail, 
1969. 


1071 


1072 


SHELL  MODE  NOISE  IN  RECIPROCATING  REFRIGERATION 
COMPRESSORS 


Ciineyt  Oztiirk  and  Aydin  Bahadir 
Tiirk  Elektrik  Endiistrisi  A.$ 

R&D  Department 

Davutpa§a,  Litres  Yolu,  Topkapi  -34020,  Istanbul,  Turkey 


ABSTRACT 

This  study  describes  the  successful  endeavor  to  understand  the  causes  of  noise 
that  appear  on  the  shell  modes  of  the  reciprocating  refrigeration  compressors. 
The  compressor  shell  is  generally  considered  as  the  acoustic  enclosure  that 
reflects  the  acoustic  energy  back  into  the  compressor  cavity  but  also  as  the 
transmitter  and  radiator  of  the  transmitted  acoustic  energy  that  could  be 
radiated  into  the  air  or  transmitted  to  the  structure.  Vibrations  of  the 
compressor  shell  can  easily  be  characterized  in  terms  of  the  modal  parameters 
that  consist  of  the  natural  frequencies,  mode  shapes  and  damping  coefficients. 
The  noise  source  harmonics  and  the  shell  resonances  couple  to  produce  the 
shell  noise  and  vibration.  The  harmonic  spacing  is  equal  to  the  basic  pumping 
frequency.  Results  of  the  studies  indicate  that  important  natural  frequencies  of 
the  compressor  shell  usually  stay  between  2000-6000  Hz  interval.  The 
important  natural  frequencies  are  first  natural  frequencies  in  the  lower  range 
with  the  longer  wavelengths  that  radiate  well. 


INTRODUCTION 

Compressor  noise  sources  are  those  processes  where  certain  portions  are 
separated  from  the  desired  energy  flow  and  transmitted  through  the  internal 
components  of  compressor  to  the  hermetic  shell  where  it  is  radiated  from  the 
shell  as  airborne  noise  on  vibration  of  supporting  structure  will  eventually 
radiate  noise  from  some  portion  of  the  structure.  Noise  sources  of  the 
reciprocating  refrigeration  compressors  can  be  classified  as  motor  noise, 
compression  process  noise  and  valve  port  flow  noise. 


1073 


In  reciprocating  compressors  there  is  very  high  density  of  noise  harmonics  even 
though  they  decay  in  amplitude  at  high  frequencies.  Generally,  these  noise 
source  harmonics  and  the  shell  resonances  couple  to  produce  shell  noise  and 
vibration 


NOISE  GENERATION  MECHANISM  OF  THE  RECIPROCATING 
REFRIGERATION  COMPRESSORS 

Significance  of  the  problem 

The  results  of  the  sound  radiation  characteristics  shown  at  figure  1  indicated 
that  certain  high  amplitude  frequency  components  had  very  distinctive  sound 
radiation  patterns.  It  was  suspected  these  frequencies  correspond  to  excitation 
of  either  structural  resonances  of  the  compressor  shell  or  acoustic  resonances  of 
the  interior  cavity  space.  Resonances  those  amplify  the  noise  and  vibration 
caused  by  pumping  harmonics  of  a  compressor  and  thus  can  be  the  cause  of 
significant  noise  problems. 


Sound  power  -  A-Wcighted 


— I - , - 1 - , - ; - P — — r — f — ! — — 'I'  "I  '"'I  — I - ”r '  i  i 

50.0  63.0  80,0  100,0  125,0  160,0  200,0  250,0  315,0  400,0  500,0  630,0  800,0  l,0k  t,3k  1,6k  2,0k  2,5k  4,0k  S,0k  63k  S,0k  10,0k 

Frequency  [Hz] 


Figure  1,  Noise  Radiation  Characteristics  of  Reciprocating  Refrigeration 
Compressor. 

Noise  sources 

Noise  in  a  compressor  is  generated  during  cyclic  compression,  discharge, 
expansion  and  suction  process.  The  character  of  noise  sources  is  harmonic  due 
to  periodic  nature  of  the  compression  process.  These  harmonics  are  present  in 
the  compression  chamber,  pressure  time  history  and  loading  of  the  compressor 
through  drive  system.  The  motor  can  not  provide  immediate  response  to  load 
harmonics  and  load  balance  is  obtained  at  the  expense  of  acceleration  and 


1074 


deceleration  of  the  motor  drive  system.  Harmonic  vibrations  of  the  motor  drive 
system  can  then  excite  the  resonant  response  in  the  compressor  components 
that  can  transmit  the  acoustic  energy  in  very  efficient  way.  The  rest  of  the  noise 
sources  are,  turbulent  nature  of  flow  depending  to  passage  through  valve  ports, 
valve  impacts  on  their  seats  and  possible  amplification  when  matched  with 
mechanical  resonances. 

MECHANICAL  FEATURES  OF  SHELL 
Mechanical  features  of  the  compressor 

The  hermetically  sealed  motor  compressor  comprises  in  general  a  motor 
compressor  unit  including  a  motor  assembly  mounted  with  a  frame  and  a  sealed 
housing  within  which  the  compressor  is  supported  by  means  of  plurality  of  coil 
springs  each  having  one  end  spring  with  the  frame  and  the  other  end  connected 
with  the  interior  of  the  housing. 

The  refrigerant  gas  as  it  is  compressed  in  the  cylinder  is  discharged  through  the 
discharge  chamber  in  the  cylinder  head  into  the  discharge  muffler.  The 
discharge  muffler  is  generally  mounted  on  the  cylinder  head  attached  in 
covering  relation  to  an  end  face  of  the  cylinder.  Where  the  sealed  casing  is 
spherical  in  shape  for  better  noise  suppression,  an  upper  end  of  the  cylinder 
head  tends  to  interfere  with  an  inner  wall  surface  of  the  casing,  a  disadvantage 
that  can  only  be  eliminated  by  increasing  the  size  of  scaled  casing  for  providing 
a  desired  hermetically  sealed  motor. 

Compressor  Shell 

The  shell  is  easily  be  characterized  with  the  modal  parameters.  The  ideal  shell 
should  be  designed  in  a  way  that  keep  all  the  excitation  frequencies  at  the  mass 
controlled  region  of  all  its  modes.  But,  depending  on  the  very  tight  constraints 
that  come  with  the  gas  dynamics  and  motor  locations,  it  is  not  allowed  to  be 
flexible  during  the  design  of  shell.  As  a  consequence  of  the  existing  design 
limitations  natural  frequencies  of  the  shell  usually  fall  between  1000-5000  Hz. 

SHELL  RADIATION 

Figure  1  illustrates  how  the  sound  pressure  level  of  a  pumping  harmonic  can 
increase  as  it  nears  a  resonant  frequency.  The  sound  pressure  level  of  the 
pumping  harmonic  increases  around  the  shell  resonances.  The  resonance 
generally  radiates  primarily  from  the  large  flat  sides  of  the  compressor.  There 
are  three  major  acoustic  cavity  resonances  400,  500  and  630  Hz  and  four  major 
structural  resonances  of  the  compressor  shell:  around  1.6  K,  2  K,  2.5  and  3.2 
K.  Hz  at  which  noise  radiates  well  in  certain  directions. 


1075 


SHELL  RESONANCES 


Figures  2,  3  and  4  illustrate  the  frequency  responses  of  the  compressor  shell 
when  measured  with  the  impact  hammer  method.  Figure  2  is  the  response  to 
the  excitation  in  x  direction,  figure  3  is  for  the  excitation  direction  and  figure  4 
is  for  the  excitation  in  z  direction. 


Figure  2,  Frequency  response  of  the  compressor  shell  when  excited  in  X 
direction. 


Figure  3,  Frequency  response  of  the  compressor  shell  when  excited  in  Y 
direction. 


1076 


Figure  4,  Frequency  response  of  the  compressor  shell  when  excited  in  Z 
direction. 


To  verify  the  hypothesis  that  resonances  were  contributing  to  some  of  the  noise 
problems  of  the  reciprocating  piston  compressor,  a  modal  analysis  of  the  shell 
and  interior  cavity  was  performed. 

For  the  shell  modal  analysis,  the  accelerometer  to  measure  the  response 
remained  stationary,  while  the  impact  location  was  moved.  The  test  was 
performed  in  this  manner  for  convenience  since  it  was  easier  to  fix  the 
accelerometer  in  one  location  and  strike  the  compressor  with  force  hammer  at 
each  grid  point  to  obtain  transfer  function  for  each  measurement  location. 
Identical  results  are  obtained  if  impact  occurred  at  a  single  point  and  the 
response  was  measured  at  each  measurement  location.  Preliminary  test  were 
performed  initially  to  identify  an  appropriate  measurement  location  at  which  all 
important  natural  frequencies  of  shell  are  detectable.  Several  force  input  and 
response  locations  were  evaluated  to  determine  the  best  location  to  mount 
accelerometer  to  measure  the  shell  response. 

The  shell  resonances  are  also  calculated  by  using  the  Structural  FEM  analysis. 
Table- 1  lists  the  natural  frequencies  predicted  in  these  studies.  During  the  finite 
element  analysis,  the  models  of  the  compressor  were  built,  based  on  the  CAD 
models.  The  shell  consists  of  7500  elements.  The  mesh  densities  are  quite 
adequate  for  the  structural  analysis  in  the  frequency  range  of  interest.  In  order 
to  investigate  the  possible  influence  of  the  crank  mechanism  on  the  natural 
frequencies  of  the  shell,  a  simple  model  of  the  crank  mechanism  was  introduced 
to  the  FE  model.  During  the  calculations,  the  crank  mechanism  was  simplified 
as  a  rigid  block  with  certain  mass  and  rotary  inertia  and  modeled  with  solid 
elements.  The  shell  and  the  crank  mechanism  have  been  suspended  with  the 


1077 


springs  from  3  positions  and  in  all  6  transitional  and  rotational  directions. 
Depending  on  the  negligible  spring  effect  on  the  longitudinal  direction, 
estimated  values  have  been  used  in  5  directions.  The  FE  model  has  been 
assumed  to  be  free-free. 


Mode 

# 

Frequency 

Hz 

Mode 

# 

Frequency 

Hz 

1 

1997 

21 

4716 

2 

22 

4925 

3 

2293 

23 

4994 

4 

24 

5 

25 

5159 

6 

26 

5454 

7 

2889 

27 

5476 

8 

3258 

28 

9 

29 

EESI^HI 

10 

30 

5783 

11 

3376 

31 

5936 

12 

3551 

32 

5999 

13 

3577 

33 

6035 

14 

34 

6055 

15 

3788 

35 

6183 

16 

3958 

36 

6237 

17 

4383 

37 

6314 

18 

4481 

38 

19 

4644 

39 

20 

4702 

40 

6701  1 

Table- 1,  Calculated  Natural  Frequencies  of  the  Shell 

Figures  5,  6,  7  and  8  illustrate  how  the  mode  shapes  of  the  shell  vary  at  the 
mode  frequencies  of  2754,  3332,  3551  and  3788  Hz  ,  These  figures  indicate 
that  the  shell  vibrates  predominantly  along  the  large  flat  sides  of  the  compressor 
at  points  where  the  suspension  springs  are  attached  to  the  shell  wall  at  these 
natural  frequencies.  When  referred  to  figure  1  of  the  noise  radiation  this 
frequency  range  is  also  the  range  where  the  noise  radiates  efficiently  from  the 
large  flat  sides  of  the  compressor.  Thus,  there  is  good  correlation  between  the 
acoustic  data  and  structural  data  for  these  frequencies.  The  slight  discrepancies 
in  the  structural  natural  frequencies  and  the  acoustic  data.  Acoustic  data  have 
been  picked  up  at  the  shell  temperature  of  the  reciprocating  piston  compressor 
that  could  reach  up  to  1 10  C. 


The  modal  analysis  results  also  indicate  that  the  compressor  suspension  springs 
are  attached  to  a  point  on  the  shell  where  the  shell  is  comparatively  compliant. 
Thus,  the  vibrational  energy  transmitted  through  the  springs  to  the  compressor 
shell  can  and  did  effectively  excite  the  shell  vibrations.  Also,  significant  shell 
vibrations  occur  along  the  large  flat  sides  of  the  compressor  shell  indicating  the 
curvature  of  the  shell  needs  to  be  increased  to  add  stifihiess  to  the  shell. 

Based  on  the  results  of  the  shell  modal  analysis,  it  is  recommended  the 
suspension  springs  moved  away  from  the  compliant  side  walls  of  the  shell.  A 
four  spring  arrangement  at  the  bottom  of  the  shell  near  corners  where  the 
curvature  is  sharp  would  reduce  the  amount  of  vibration  energy  transferred  to 
the  shell  because  of  the  reduced  input  mobility  of  the  shell  at  these  locations. 

It  is  also  believed  increasing  the  stiffness  of  the  shell  by  increasing  the  curvature 
will  provide  noise  reduction  benefits.  The  greater  shell  stiffness  lowers  the 
amplitude  of  the  shell  vibrations.  Figure  9,  illustrate  the  third  octave  change  in 
compressor  noise  with  the  same  compressor  in  the  new  shell.  An  over  all  noise 
level  of  5  dBA  has  been  obtained. 


Figure  9  Compressor  noise  level  improvement  after  the  shell  modification. 

The  increased  shell  stiffness  also  raises  the  natural  frequencies  of  the  shell 
where  there  is  less  energy  for  transfer  function  response.  However,  there  is  a 
possible  disadvantage  to  increasing  stiffness  of  the  shell.  The  higher  natural 
frequency  lowers  critical  frequency  of  the  shell  thus  reducing  transmission  loss 
of  the  shell. 


1081 


Damping  treatments  can  also  have  obvious  benefits  in  vibration  and  noise 
reduction.  Visco  elastic  and  Acoustic  dampings  are  considered  to  avoid  the 
shell  excitations.  The  application  of  dampers  can  also  provide  up  to  5  dBA 
reductions  when  appropriately  located  on  the  shell. 

CONCLUSION 

The  results  of  studies  indicate  that  structural  resonances  of  the  shell  are  indeed 
amplifying  the  noise  due  to  the  pumping  harmonics  of  the  reciprocating 
refrigeration  compressor  to  cause  significant  noise  radiation  outside  of 
compressor. 

In  order  to  tackle  with  this  noise  problem,  within  the  scope  of  these  studies 
two  different  effective  shell  noise  control  are  considered  based  on  the  results  of 
numerical  and  experimental  structural  analysis  and  acoustic  features  of  the 
reciprocating  compressor.  First,  shell  noise  control  method  is  the  redesign  of 
the  shell  with  increased  stiffness  by  replacing  all  the  abrupt  changes  in  the 
curvature  with  the  smooth  continuous  changes.  It  is  apparent  that  change  in  the 
shell  configuration  can  shift  the  first  shell  resonance  from  1750  Hz  to  nearly 
3200  Hz.  The  results  of  the  redesign  effects  can  reach  up  to  3-5  dBA  reduction 
on  third  octave  noise  levels.  Second  treatment  that  could  be  applied  against  the 
excitation  of  shell  resonances  are  considered  as  the  acoustic  and  viscoelastic 
dampers.  These  dampers  can  be  chosen  to  operate  efficiently  at  the  shell 
resonant  frequencies.  These  two  applications  can  also  provide  up  to  2-5  dBA 
reduction  on  the  third  octave  band  of  the  noise  emission  but  the  long  term 
endurance  and  temperature  dependence  of  these  materials  can  always  be  a 
question  mark  when  considered  from  the  manufacturer  point  of  view. 


REFERENCES 

1.  JFROBATTAand  ID.  JONES  1991,  Purdue  University,  School  of 
Mechanical  Engineering,  The  Ray  Herrick  Laboratories,  Report  no :  1912-1 
HL  91-9P,  73-84,  Investigation  of  Noise  Generation  Mechanisms  and 
Transmission  Paths  of  Fractional  Horsepower  Reciprocating  Piston  and 
Rolling  Piston  Compressors 

2.  HAMILTON  I F  1988,  Purdue  University ,  School  of  Mechanical 
Engineering,  The  Ray  Herrick  Laboratories,  207-213  Measurement  and 
Control  of  Compressor  Noise 

3.  C  OZTURK,  A  AQIKGOZ  and  J  L  MIGEOT  1996,  International 
Compressor  Engineering  Conference  at  Purdue,  Conference  Proeceeding , 
Volume  II,  697-703,  Radiation  Analysis  of  the  Reciprocating  Refrigeration 
Compressor  Casing 


1082 


A  COMPARATIVE  STUDY  OF  MOVING  FORCE  IDENTIFICATION 


T.H.T.  Chan,  S.S.  Law,  T.H.  Yung 

Department  of  Civil  &  Structural  Engineering, 

The  Hong  Kong  Polytechnic  University,  Hung  Horn,  Kowloon,  Hong  Kong 


ABSTRACT 

Traditional  ways  to  acquire  truck  axle  and  gross  weight  information  are 
expensive  and  subject  to  bias,  and  this  has  led  to  the  development  of  Weigh- 
in-Motion  (WIM)  techniques.  Most  of  the  existing  WIM  systems  have  been 
developed  to  measure  only  the  static  axle  loads.  However  dynamic  axle  loads 
are  also  important.  Some  systems  use  instrumented  vehicles  to  measure 
dynamic  axle  loads,  but  are  subject  to  bias.  These  all  prompt  the  need  to 
develop  a  system  to  measure  the  dynamic  axle  loads  using  an  unbiased 
random  sample  of  vehicles.  This  paper  aims  to  introduce  four  methods  in 
determining  such  dynamic  forces  from  bridge  responses.  The  four  methods  are 
compared  with  one  another  based  on  maximum  number  of  forces  to  be 
identified,  minimum  number  of  sensors,  sensitivity  towards  noise  and  the 
computation  time.  It  is  concluded  that  acceptable  estimates  could  be  obtained 
by  all  the  four  methods.  Further  work  includes  merging  the  four  methods  into 
a  Moving  Force  Identification  System  (MFIS). 


INTRODUCTION 

The  truck  axle  and  gross  weight  information  have  application  in  areas 
such  as  the  structural  and  maintenance  requirements  of  bridges  and  pavements. 
However,  the  traditional  ways  to  acquire  that  are  expensive  and  subject  to  bias, 
and  this  has  led  to  the  development  of  Weigh-in-Motion  (WIM)  techniques. 
Some  systems  are  road-surface  systems  which  make  use  of  piezo-electric 
(pressure  electricity)  or  capacitive  properties  to  develop  a  plastic  mat  or 
capacitive  sensors  to  measure  axle  weight  [1].  Another  kind  of  WIM  system  is 
the  under-structure  systems  in  which  sensors  are  installed  under  a  bridge  or  a 
culvert  and  the  axle  loads  are  computed  from  the  measured  responses  e.g. 
AXWAY  [2]  and  CULWAY  [3].  All  the  above  mentioned  systems  can  only 
give  the  equivalent  static  axle  loads.  However  dynamic  axle  loads  are  also 
important  as  they  may  increase  road  surface  damage  by  a  factor  of  2  to  4  over 
that  caused  by  static  loads  [4].  Some  systems  use  instrumented  vehicles  to 
measure  dynamic  axle  loads  [5],  but  are  subject  to  bias.  These  all  prompt  the 
need  to  develop  a  system  to  measure  the  dynamic  axle  loads  using  unbiased 
random  samples  of  vehicles.  Four  methods  are  developed  to  determine  such 


1083 


dynamic  forces  from  bridge  responses  which  include  bending  moments  or 
accelerations. 


EQUATION  OF  MOTIONS  FOR  MOVING  LOADS 

The  moving  force  identification  methods  described  in  this  paper  are  the 
inverse  problems  of  an  predictive  analysis  which  is  defined  by  0‘ Connor  and 
Chan  [6]  as  an  analysis  to  simulate  the  structural  response  caused  by  a  set  of 
time-varying  forces  running  across  a  bridge.  Two  models  can  be  used  for  this 
kind  of  analysis. 


A.  Beam-Elements  Model 


0‘ Connor  and  Chan  [6]  model  the  bridge  as  an  assembly  of  lumped 
masses  interconnected  by  massless  elastic  beam  elements  as  shown  in  Figure  1, 
and  the  nodal  responses  for  displacement  or  bending  moments  at  any  instant 
are  given  by  Equations  (1)  and  (2)  respectively. 


Moving  Loads 


...  ip.  Ip,  ^ 

- D — D — 2^ 


Lumped  Masses  1  2  ...  N-1  N 


Figure  1  -  Beam-Elements  Model 


{Y}  =  [Y,]{F}-lY,][Am]{n-m[C]{Y}  (1) 

{5M}  =  [5MJ{P}-[5M;][Am]{y}-[5MJ[C]{f}  (2) 

where  [P]  is  a  vector  of  wheel  loads,  [Am]  is  a  diagonal  matrix  containing 

values  of  lumped  mass,  [C]  is  the  damping  matrix,  5M,L,7,7are  the  nodal 
bending  moments,  displacements,  velocities  and  accelerations  respectively, 
[R,^]  {'R  ’  can  be  Y  or  BAP)  is  an  m  x  n  matrix  with  the  ith  column  representing 
the  nodal  responses  caused  by  a  unit  load  acting  at  the  position  of  the  ith  wheel 
load  and  [i?J  {'R'  can  be  Y  or  BM)  is  an  n  x  n  matrix  with  the  ith  column 
representing  the  nodal  responses  caused  by  a  unit  load  acting  at  the  position  of 
ith  internal  node. 


1084 


B.  Continuous  Beam  Model 


Assuming  the  beam  is  of  constant  cross-section  with  constant  mass  per 
unit  length,  having  linear,  viscous  proportional  damping  and  with  small 
deflections,  neglecting  the  effects  of  shear  deformation  and  rotary  inertia 
(Bernoulli-Euler’s  beam),  and  the  force  is  moving  from  left  to  right  at  a 
constant  speed  c,  as  shown  in  Figure  2,  then  the  equation  of  motion  can  be 
written  as 


P 


d-v{xj) 

a- 


a 


+  EI 


^v(x,0 

a- 


S(x-ct)f{t) 


(3) 


where  v(x,0  is  the  beam  deflection  at  point  x  and  time  t;  p  is  mass  per  unit 
length;  C  is  viscous  damping  parameter;  £  is  Young's  modulus  of  material;  I 
is  the  second  moment  of  inertia  of  the  beam  cross-section;  f(t)  is  the  time- 
varying  force  moving  at  a  constant  speed  of  c,  and  Sft)  is  Dirac  delta  function. 


\/(0 


— O 


Figure  2  -  Simply  supported  beam  subjected  to  a  moving  force  f(t) 

Based  on  modal  superposition,  the  dynamic  deflection  y(x,t)  can  be 
described  as  follows; 


v{x,t)  =  Y,<i'„{x)q„{t) 

}}=\ 


(4) 


where  n  is  the  mode  number;  0n(x^)  is  the  mode  shape  function  of  the  n-th 
mode  and  qyi(t)  is  the  n-th  modal  amplitudes. 

Based  on  the  above  mentioned  predictive  analyses,  four  Moving  Force 
Identification  Methods  (MFIM)  are  developed. 


1085 


FIRST  MOVING  FORCE  IDENTIFICATION  METHOD 
INTERPRETIVE  METHOD  I  -  BEAM-ELEMENTS  MODEL  (IMI) 


It  is  an  inverse  problem  of  the  predictive  analysis  using  beam-elements 
model  From  Equation  (1),  it  can  be  seen  that  if  Y  is  known  at  all  times  for  all 
interior  nodes,  then  7  and  Y  can  be  obtained  by  numerical  differentiation. 
Equation  (1)  becomes  an  overdetermined  set  of  linear  simultaneous  equations 
in  which  the  P  may  be  solved  for  them.  However  a  particular  difficulty  arises 
if  measured  BM  are  used  as  input  data.  Remembering  that  the  moving  loads  P 
are  not  normally  at  the  nodes,  the  relation  between  nodal  displacements  and 
nodal  bending  moments  is 

{r}^[Y,]{BM}  +  [Yc]{B}  (5) 

where  allows  for  the  deflections  due  to  the  additional  triangular 

bending  moment  diagrams  that  occur  within  elements  carrying  one  or  more 
point  loads  P.  [}^]can  be  calculated  from  the  known  locations  of  the  loads. 
[}^^]  and  {BM}aic  known,  but  {7}  cannot  be  determined  without  a 
knowledge  of  (P}.  0‘Connor  and  Chan  [6]  describe  a  solution  uses  values  of 
{P}  assumed  from  the  previous  time  steps. 


SECOND  MOVING  FORCE  IDENTIFICATION  METHOD 
INTERPRETIVE  METHOD  II  -  CONTINUOUS  BEAM  MODEL  (IMII) 

From  the  predictive  analysis  using  continuous  beam  model,  if  the  ith- 

iTTX 

mode  shape  function  of  the  simply  supported  Euler‘s  beam  is  sin-—-,  then 
the  solution  of  Equation  (3)  takes  the  form 

v  =  ^sin— F;(0  (6) 

/=i  L 

where  V.(t) ,  (i  =  1,  2,  •  •  •)  are  the  modal  displacements. 

Substitute  Equation  (6)  into  Equation  (3),  and  multiply  each  term  of 
Equation  (3)  by  the  mode  shape  function  sin(y;r.x  /  L) ,  and  then  integrate  the 
resultant  equation  with  respect  to  x  between  0  and  L  and  use  the  boundary 
conditions  and  the  properties  of  Dirac  function.  Consequently,  the  following 
equation  can  be  obtained 


1086 


op 


u  2  El  ^ 

where  =  —5 — -  C(y,  = 


C 


2/rryy, 


at  the  j-th  mode. 


If  there  are  more  than  one  moving  loads  on  the  beam,  Equation  (7)  can  be 
written  as 


.  7r(ct-Xk) 

._7r(ct~x,) 

7r{ct-x^)  ■ 

■^.1 

'2CM' 

'cojv; 

sin - ^ 

L 

sin - ^  • 

L 

sin - — 

L 

'K 

V, 

2C,co,K 

+ 

CO  IVj 

_  2 

.  27r(ct-Xt) 
sin 

.  27r(ct~x^} 
sm - =-  ■ 

.  27r{ct  -  x. ) 

••  sm — - — 

Pi 

~  Ml 

A 

.1 

_2C,.a)/„_ 

_coX_ 

.  n7r(ct-x,) 
sin — ^ ^ 

.  riKict-x^) 
sin — ^ - —  . 

.  n7c{ct-x^) 

••  sin — - — 

A. 

L 

L 

L 

(8) 


in  which  is  the  distance  between  the  k-th  load  and  the  first  load  and  x,  =  0 . 


If  Pi  Pk  are  constants,  the  closed  form  solution  of  Equation  (3)  is 

/  N  ^  1  •  ■  j7E{ct-Xi)  a  .  .  .  ,  ^  {Q\ 

48E/f:i'  L  \  L  j  ^  J 

nc 


in  which  a  = 


Lcd 


If  we  know  the  displacements  of  the  beam  at  x, ,  x, ,  •  •  • ,  ,  the  moving 

loads  on  the  beam  are  given  by 

{v}  =  [S, „.]{/>}  (10) 

in  which  {v}=[v|  v,  ■■■  v,]'  {•^’}  =  [-^1  ^2  ■" 


^n- 


,  where 


L"  A  1  .  j7rx,J  .  J7r{ct-x.)  a  .  ,  .  ,  , 

=  7577X^2772 - 27^“ “7~  - 7 - /c) 

48£:/^;-0- -a  )  L  \  L  j 


1087 


If  l>k,  that  means  the  number  of  nodal  displacements  is  larger  than  or 
equal  to  the  number  of  axle  loads,  then  according  to  the  least  squares  method, 
the  equivalent  static  axle  load  can  be  given  by 

{/>}=([s,,]''[s„])''[s,q^{v}  (11) 

If  the  loads  are  not  constant  with  time,  then  central  difference  is  used  to 
proceed  from  modal  displacements  to  modal  velocities  and  accelerations. 
Equation  (8)  becomes  a  set  of  linear  equations  in  which  P^.  for  any  instant  can 
be  solved  by  least  squares  method.  Similar  sets  of  equations  could  be  obtained 
for  using  bending  moments  to  identify  the  moving  loads. 


THIRD  MOVING  FORCE  IDENTIFICATION  METHOD 
TIME  DOMAIN  METHOD  (TDM) 


This  method  is  based  on  the  system  identification  theory  [7].  Substituting 
Equation  (4)  into  Equation  (3),  and  multiplying  each  term  by  0j(x),  integrating 
with  respect  to  x  between  0  and  L,  and  applying  the  orthogonality  conditions, 
then 


dt- 


dt 


M., 


(12) 


where  con  is  the  modal  frequency  of  the  n-th  mode;  is  the  damping  ratio  of 
the  n-th  mode;  Mn  is  the  modal  mass  of  the  n-th  modt,  pn(t)  is  the  modal  force 
and  the  mode  shape  function  can  be  assumed  as  0„(x)  =  sin(^;w  /  L) . 

Equation  (12)  can  be  solved  in  the  time  domain  by  the  convolution 
integral,  and  yields 

=  (13) 

^11  0 

where  ~  sin(^u„0,  t  >0  (14) 

and  co\,  (15) 

Substituting  Equation  (13)  into  Equation  (4),  the  dynamic  deflection  of 
the  beam  at  point  x  and  time  t  can  be  found  as 

v(x,0  =  y — ^sin^^  sin6>|,(^  -  '^)sin^^^^/(T)dr  (16) 

fxpLo),,  L  i  L 


1088 


A.  Force  Identification  from  Bending  Moments 


The  bending  moment  of  the  beam  at  point  x  and  time  t  is 


m{xj)  =  -El - 


3c- 


(17) 


Substituting  Equation  (16)  into  Equation  (17),  and  assuming  the  force  f(t) 
is  a  step  function  in  a  small  time  interval  and  f(t)  =0  at  the  entry  and  exit,  then 
let 


^  2El7r'  n~  .  yitu:  ^ 

C,„  =  — 73 - ^sm— Ar, 

pi  co„  L 

{k)  =  sin(<:y'„  A/A:), 

^  .  ,n7(cNi 

52(*)  =  sm(— ^/c) 


(18) 


(19) 


N„=- 

Equation  (17)  can  be  expressed  as 


cEt 


m(2)] 

0 

...  0  ' 

/(i) 

m(3) 

< 

■=S  c,„ 
/)  =  1 

E;,S,{2)S,i\) 

e:S,{\)S,{2) 

...  0 

/(2) 

m{N) 

_E::"S,iN-\)S,il) 

E;^--S,{N~2)S,i2) 

Kc. 

(20) 


where  A^  is  the  sample  interval  and  N+I  is  the  number  of  sample  points,  and 


f>„=£r'''“*'S,(Al-A^„+l)S,(//„-l) 


Equation  (20)  can  be  simplified  as 

B  f  =  m  (21) 

(A'-l)x(;V/^-|)  ('V/^->)xl  (,V_l)xl 

U  N  =  matrix  B  is  a  lower  triangular  matrix.  We  can  directly  find 
the  force  vector  f  by  solving  Equation  (21).  If  77  >  and/or  77/  bending 
moments  (77/  >  1)  are  measured,  least  squares  method  can  be  used  to  find 

the  force  vector  f  from 


1089 


(22) 


B,  ■ 

m, 

f 

m. 

®-v,. 

The  above  procedure  is  derived  for  single  force  identification.  Equation 
(21)  can  be  modified  for  two-forces  identification  using  the  linear 
superposition  principle  as 

“B.  0 

B,  B, 

B.  B, 

where  B 3  [N^x  (Nb-1)]  ,  Bt  [(N  -1-2NJx  (Ng-l)],  and  B^  [NjX  (Nb  -1)]  are 
sub-matrices  of  matrix  B.  The  first  row  of  sub-matrices  in  the  first  matrix 
describes  the  state  having  the  first  force  on  beam  after  its  entry.  The  second 
and  third  rows  of  sub-matrices  describe  the  states  having  two-forces  on  beam 
and  one  force  on  beam  after  the  exit  of  the  first  force. 

B.  Identification  from  Bending  Moments  and  Accelerations 


Similarly  the  acceleration  response  of  the  beam  can  be  expressed  as 

A  f  =  V 


(24) 


The  force  can  also  be  found  from  the  measured  acceleration  from 
Equation  (24).  If  the  bending  moments  and  accelerations  responses  are 
measured  at  the  same  time,  both  of  them  can  be  used  together  to  identify  the 
moving  force.  The  vector  m  in  Equation  (21)  and  v  in  Equation  (24)  should  be 
scaled  to  have  dimensionless  unit,  and  the  two  equations  are  then  combined 
together  to  give 


A/iv||J  [v/llvlj 


(25) 


where  Ihli  is  the  norm  of  the  vector. 


FOURTH  MOVING  FORCE  IDENTIFICATION  METHOD 
FREQUENCY  DOMAIN  METHOD  (FDM) 

Equation  (12)  can  also  be  solved  in  the  Frequency  Domain.  Performing 
the  Fourier  Transform  for  Equation  (12), 


1090 


1 


1 


(26) 


where 


co;,  -CO-  +24„co„co  M„ 


W 


i 

—00 


(27) 

(28) 


Let 


_ 1 _ 

co;,-co-  +2^„a„a 


(29) 


Hn(co)  is  the  frequency  response  function  of  the  n-th  mode.  Performing  the 
Fourier  Transform  of  Equation  (4),  and  substituting  Equations  (26)  and  (29) 
into  the  resultant  equation,  the  Fourier  Transform  of  the  dynamic  deflection 
v(x,t)  is  obtained  as 


A.  Force  Identification  from  Accelerations 

Based  on  Equation  (30),  the  Fourier  Transform  of  the  acceleration  of  the 
beam  at  point  .x  and  time  t  can  be  written  as 

V(x,a)  =  -®^X^(D„(x)//„(cy)P„(cy)  (31) 

Considering  the  periodic  property  of  the  Discrete  Fourier  Transform  (DFT), 
and  let 


A/^m- 


Equation  (32)  can  be  rewritten  as 


(32) 


1091 


,m  =  0,l,  (33) 


V{m)  =  X  H„(m)'i'„(m)[F,{0)  +  iF,(0)] 

/;=1 

A^/2-1  «  _ 
k=\  H=t 

Nn~\  =0  _ 

/t  =  l  /)  =  l 

+E  /2)[F„(A^  /2)  -  iivCA^  /  2)] 

«=I 

where  is  the  Fourier  Transform  of  the  n-th  mode  shape,  and  F  is  the 
Fourier  Transform  of  the  moving  force. 

Writing  Equation  (33)  into  matrix  form  and  dividing  F  and  V  into  real  and 
imaginary  parts,  it  yields 


Because  F,  (0)  =  0,  Fj  (N  /  2)  =  0,Vj  (0)  =  0,F;  ( /  2)  =  0 ,  Equation  (34)  can 
be  condensed  into  a  set  of  N  order  simultaneously  equations  as 

(35) 

Fr  and  F,  can  be  found  from  Equation  (35)  by  solving  the  Nth  order  linear 
equation.  The  time  history  of  the  moving  force  f(t)  can  then  be  obtained  by 
performing  the  inverse  Fourier  Transformation. 

If  the  DFTs  are  expressed  in  matrix  form,  the  Fourier  Transform  of  the 
force  vector  f  can  be  written  as  follows  if  the  terms  in  f  are  real  [8]. 

F  =  — Wf  (36) 

N 

whereW  =  e''^‘‘^^^  (37) 


1092 


0  0  0  0  0 

0  1  2  •••  -2  -1 

0  2  4  •••  -4  -2 

0  -2  -4  •••  4  2 

0  -1  -2  2  1 


yv  X 


The  matrix  W  is  an  unitary  matrix,  which  means 

w-‘  =(w*  y 


(38) 


where  W*  is  a  conjugate  of  W.  Substituting  Equation  (36)  into  Equation  (35), 


V  =  — Al 

N 


W  0 

- 1 

X 

_ Ij 

(39) 


or 


V=^  A  fe 


(40) 


linking  the  Fourier  Transform  of  acceleration  V  with  the  force  vector  fg  of 
the  moving  forces  in  the  time  domain.  Wg  is  the  sub-matrix  of  W.  If  N  = 
fg  can  be  found  by  solving  the  Mh  order  linear  equations.  If  N>  or  more 
than  one  accelerations  are  measured,  the  least  squares  method  can  be  used  to 
find  the  time  history  of  the  moving  force  f(t). 

Equation  (40)  can  be  rewritten  as  follows 


(41) 


relating  the  accelerations  and  force  vectors  in  the  time  domain.  Also  if  N  - 
Nb,  fe  can  be  found  by  solving  the  Nth  order  linear  equation.  If  N  >  Nb  or 
more  than  one  acceleration  are  measured,  the  least  squares  method  can  be 
used  to  find  the  time  history  of  the  moving  force  f(t). 


If  only  Nc  {Nc  ^  N)  response  data  points  of  the  beam  are  used,  the 
equations  for  these  data  points  in  Equation  (41)  are  extracted,  and  described  as 


V,  =(w*)"  A 

C  V  B  /  A/x/V 


c 


B 

Nr  X  N 


W3 

/Vx;V, 


yV„x! 


(42) 


1093 


In  usual  cases  Nc  >  Ng,  so  the  least  squares  method  is  used  to  find  the  time 
history  of  the  moving  force  f(t).  More  than  one  acceleration  measurements  at 
different  locations  can  be  used  together  to  identify  a  single  moving  force  for 
higher  accuracy. 


B.  Identification  from  Bending  Moments  and  Accelerations 


Similarly,  the  relationships  between  bending  moment  m  (and  M )  and 
the  moving  force  f  can  be  described  as  follows, 

M  =  ^  B  W  is  (43) 

/Vxl  N  N^Nn 


(44) 

(45) 


The  force  vector  fg  can  be  obtained  from  the  above  three  sets  of 
equations.  Furthermore,  these  equations  can  be  combined  with  Equations  (40), 
(41)  and  (42)  to  construct  overdetermined  equations  before  the  equations  are 
scaled.  Two  forces  identification  are  developed  using  the  similar  procedure  as 
that  for  the  Time  Domain  Method. 


COMPARATIVE  STUDY 

The  first  moving  force  identification  method  is  implemented  in  a 
computer  program  using  FORTRAN,  while  the  other  three  methods  are 
implemented  under  the  environment  of  a  high  performance  numerical 
computation  and  visualization  software.  The  predictive  analysis  using  beam- 
elements  model  is  used  to  generate  the  theoretical  bridge  responses  and  the 
four  moving  force  identification  methods  then  use  these  responses  to  recover 
the  original  dynamic  loads.  In  this  study,  if  at  least  80%  of  the  identified 
forces  at  any  instant  of  any  load  lie  within  ±  1 0%  of  the  original  input  force, 
the  method  is  considered  acceptable.  It  is  found  that  all  the  four  methods  can 
give  acceptable  results. 

It  is  decided  to  carry  out  a  preliminary  comparative  study  on  the  four 
methods  in  order  to  study  the  merits  and  limitations  of  each  method  so  as  to 
consider  the  future  development  of  each  method  and  devise  a  plan  to  develop  a 


1094 


moving  force  identification  system  which  can  make  use  of  the  benefits  of  all 
the  four  methods. 


A.  Maximum  Number  of  Forces 


This  is  to  examine  the  maximum  number  of  axle  loads  that  can  be 
identified  by  each  method.  Theoretically,  provided  that  sufficient  number  of 
nodal  sensors  are  installed,  IMI  and  IMII  can  be  used  to  identify  as  many  loads 
as  the  system  allows.  Basically,  the  number  of  axle  loads  cannot  be  larger  than 
the  number  of  nodal  sensors.  Regarding  TDM  and  FDM,  as  the  formulation  of 
the  governing  equation  is  derived  for  two  moving  forces,  the  maximum 
number  of  axle  loads  that  can  be  identified  is  two. 

B.  Minimum  Number  of  Sensors 


Based  on  a  study  of  common  axle  spacings  of  vehicles  currently 
operating  on  Australian  roads,  and  the  cases  with  zero  nodal  responses, 
0‘ Connor  and  Chan  [6]  state  the  relationships  of  the  minimum  number  of 
sensors  used  for  IMI  and  the  span  length  of  a  bridge  as  follows: 


Using  bending  moment,  for  span  length  L  >  4.8m, 

Min.  number  of  nodal  moments  required  =  int{ 
Using  displacements,  for  span  length  L  >  13.8m, 

Min.  number  of  nodal  displacement  required  =  INj{ 
and  for  span  length  L  <13.8m, 

Min.  number  of  nodal  displacement  required 


L  -  4.8' 
1.7 

L-UX 


3.7 


+  4 


+  6 


(46) 

(47) 

(48) 


For  IMII,  it  is  found  that  the  number  of  sensors  required  are  generally  less 
than  that  for  IMI.  Regarding  TDM  and  FDM,  the  programs  are  not  as  flexible 
as  that  for  IMI  and  IMII  and  it  is  not  easy  to  change  the  number  of  sensors. 
Meanwhile  the  sensors  are  fixed  to  be  at  1/4,  1/2  and  3/4  of  the  span. 

C.  Sensitivity  towards  Noise 


In  general,  all  the  four  methods  can  compute  the  identified  forces  exactly 
the  same  as  those  given  to  the  predictive  analysis  to  generate  the 
corresponding  responses.  It  is  decided  to  add  white  noise  to  the  calculated 
responses  to  simulate  polluted  measurements  and  to  check  their  sensitivity 
towards  noise.  The  polluted  measurements  are  generated  by  the  following 


1095 


equations: 


m  =  n>o.nkui«.«]  x  N„, 


(49) 


where  Ep  is  a  specified  error  level;  is  a  standard  normal  distribution 
vector  (with  zero  mean  value  and  unity  standard  deviation). 

Several  cases  are  studied  using  Ep  =1%,  3%,  5%  and  10%.  It  is  found 
that  when  using  bending  moments  for  IMI  and  IMII,  and  if  Ep  is  less  than  3%, 
acceptable  results  can  be  obtained.  For  noise  which  is  greater  than  3%,  a 
smoothing  scheme  should  be  adopted  to  smooth  the  simulated  data. 
Acceptable  results  cannot  be  obtained  for  Ep  >  10%.  Besides,  both  IMI  and 
IMII  cannot  give  acceptable  results  when  using  displacements. 

Both  TDM  and  FDM  cannot  give  acceptable  results  when  using 
displacements  only,  accelerations  only  or  bending  moments  only.  In  general 
TDM  and  FDM  are  less  sensitive  to  noise  when  comparing  to  IMI  and  IMII. 
They  can  give  acceptable  results  for  Ep  up  to  5  %  without  any  smoothing  of 
the  polluted  simulated  data. 

D.  Computation  Time 

In  general,  the  computer  program  for  IMI  only  takes  few  seconds  to 
identify  moving  forces.  In  order  to  compare  the  computation  time,  IMI  is 
implemented  in  the  same  environment  as  the  other  three  methods.  It  is  found 
that  IMI  and  IMII  take  about  2-3  minutes  to  give  the  identified  forces  for  a 
case  of  two  axle  loads  using  a  80486  computer.  However,  under  the  same 
working  conditions,  TDM  and  FDM  almost  take  a  whole  day  for  any  one  of 
them  to  identify  two  moving  forces.  It  is  due  to  the  fact  that  both  of  them 
require  to  set  up  an  huge  parametric  matrix. 


CONCLUSIONS 

Four  methods  are  developed  to  identify  moving  time-varying  force  and 
they  all  can  produce  acceptable  results.  From  a  preliminary  comparative  study 
of  the  methods,  it  is  found  that  IMI  and  IMII  have  a  wider  applicability  as  the 
locations  of  sensors  are  not  fixed  and  it  can  identify  more  than  two  moving 
forces.  However,  TDM  and  FDM  are  less  sensitive  to  noise  and  require  less 
number  of  sensors.  It  is  decided  to  further  improve  the  four  methods  and  then 
a  more  detailed  and  systematic  comparison  can  be  carried  out  afterwards.  The 
possible  development  of  the  methods  are  described  as  follows. 


1096 


Both  the  IMI  and  IMII  are  developed  to  work  with  one  kind  of  responses, 
e.g.  either  displacements  or  bending  moments.  It  is  suggested  to  modify  the 
programs  to  use  mixed  input  parameter,  e.g.  use  bending  moments  as  well  as 
accelerations  as  that  for  TDM  or  FDM.  Regarding  the  TDM  and  FDM,  as  the 
basic  formulations  are  based  on  two-axle  moving  forces,  so  it  is  necessary  to 
modify  the  governing  equations  for  multi-axle.  In  addition,  the  computation 
time  for  TDM  or  FDM  under  the  environment  of  the  high  performance 
numerical  computation  and  visualization  software  used  is  unbearable.  It  is 
expected  that  the  time  will  be  significantly  reduced  if  the  methods  are 
implemented  in  programs  using  standard  programming  languages  like 
FORTRAN  90  or  C.  Then  the  four  methods  can  be  combined  together  and 
merged  into  a  Moving  Force  Identification  System  (MFIS)  so  that  it  can 
automatically  select  the  best  solution  routines  for  the  identification. 


ACKNOWLEDGMENT 

The  present  project  is  funded  by  the  Hong  Kong  Research  Grants  Council. 


REFERENCES 

1.  Davis,  P.  and  Sommerville,  F.,  Low-Cost  Axle  Load  Determination, 
Proceedings,  13th  ARRB  &  5th  REAAA  Combined  Conference,  1986, 
Part  6,  p  142-149. 

2.  Peters,  R.J.,  AXWAY  -  a  System  to  Obtain  Vehicle  Axle  Weights, 
Proceedings,  12th  ARRB  Conference,  1984,  12  (2),  p  10-18. 

3.  Peters,  R.J.,  CULWAY  -  an  Unmanned  and  Undetectable  Highway  Speed 
Vehicle  Weighing  System,  Proceedings,  13th  ARRB  &  5th  REAAA 
Combined  Conference,  1986,  Part  6,  p  70-83. 

4.  Cebon,  D.  Assessment  of  the  Dynamic  Wheel  Forces  Generated  by 
Heavy  Vehicle  Road  Vehicles.  Symposium  on  Heavy  Vehicle  Suspension 
Characteristics,  ARRB,  1987. 

5.  Cantineni,  R.,  Dynamic  Behaviour  of  Highway  Bridges  Under  The 
Passage  of  Heavy  Vehicles.  Swiss  Federal  Laboratories  for  Materials 
Testing  and  Research  (EMPA)  Report  No.  220,  1992,  240p. 

6.  O'Connor,  C.  and  Chan,  T.H.T.,  Dynamic  Wheel  Loads  from  Bridge 
Strains.  Structural  Engineering  ASCE,  1 14  (STS),  1988,  p.  1703- 1723. 

7.  Briggs,  J.C.  and  Tse,  M.K.  Impact  Force  Identification  using  Extracted 
Modal  Parameters  and  Pattern  Matching,  International  Journal  of  Impact 
Engineering,  1992,  Vol.  12,  p361-372. 

8.  Bendat,  J.S.  and  Piersol,  J.S.,  Engineering  Application  of  Correlation  and 
Spectral  Analysis.  John  Wiley  &  Sons,  Inc.  Second  Edition,  1993. 


1097 


1098 


ESTIMATING  THE  BEHAVIOUR  OF  A 
NONLINEAR  EXPERIMENTAL  MULTI  DEGREE 
OF  FREEDOM  SYSTEM  USING  A  FORCE 
APPROPRIATION  APPROACH 


P.A.  Atkins  J.R. Wright 

Dynamics  and  Control  Research  Group 
School  of  Engineering,  Simon  Building,  University  of  Manchester, 
Oxford  Road,  Manchester.  M13  9PL 


ABSTRACT 

The  identification  of  nonlinear  multi  degree  of  freedom  systems  involves  a 
significant  number  of  nonlinear  cross  coupling  terms,  whether  the  identifi¬ 
cation  is  carried  out  in  spatial  or  modal  domains.  One  possible  approach 
to  reducing  the  order  of  each  identification  required  is  to  use  a  suitable 
pattern  of  forces  to  drive  any  mode  of  interest.  For  a  linear  system,  the 
force  pattern  required  to  drive  a  single  mode  is  derived  using  a  Force  Ap¬ 
propriation  method.  This  paper  presents  a  method  for  determining  the 
force  pattern  necessary  to  drive  a  mode  of  interest  of  a  nonlinear  system 
into  the  nonlinear  region  whilst  the  response  is  controlled  to  remain  in  pro¬ 
portion  to  the  linear  mode  shape.  Such  an  approach  then  allows  the  direct 
nonlinear  modal  terms  for  that  mode  to  be  identified  using  the  Restoring 
Force  method.  The  method  for  determining  the  relevant  force  patterns  is 
discussed.  The  implementation  of  the  method  for  experimental  systems  is 
considered  and  experimental  results  from  a  two  degree  of  freedom  ’bench¬ 
mark  structure’  are  presented. 


INTRODUCTION 

Force  Appropriation  [1]  is  used  in  the  analysis  of  linear  systems  to  de¬ 
termine  the  force  patterns  which  will  induce  single  mode  behaviour  when 
applied  at  the  relevant  natural  frequency.  This  technique  is  used  in  the 
aerospace  industry  during  Ground  Vibration  Tests:  each  normal  mode  of 


1099 


a  structure  is  excited  using  the  derived  force  pattern  and  thus  identified 
in  isolation.  Current  practice,  when  the  presence  of  nonlinearity  is  sus¬ 
pected,  is  to  increase  input  force  levels  and  monitor  the  variation  of  tuned 
frequencies.  Some  information  about  the  type  of  nonlinearity  present  may 
be  found,  but  no  analytical  model  can  be  derived.  Thus  predictions  for 
behaviour  at  higher  levels  of  excitation  axe  not  possible. 

A  number  of  techniques  for  identifying  nonlinearity,  for  example  the  Restor¬ 
ing  Force  method  [2],  have  been  demonstrated  on  systems  with  low  num¬ 
bers  of  degrees  of  freedom.  Unfortunately,  in  practice,  structures  have  a 
large  number  of  degrees  of  freedom,  often  with  a  high  modal  density.  A 
classical  Restoring  Force  approach  to  the  identification  of  such  systems 
could  involve  a  prohibitive  number  of  cross  coupling  terms.  The  ability  to 
treat  each  mode  separately,  by  eliminating  the  effects  of  the  cross  coupling 
terms,  would  thus.be  advantageous.  Subsequent  tests  could  then  evaluate 
the  cross  coupling  terms. 

For  these  reasons  it  would  be  useful  to  extend  Force  Appropriation  to  the 
identification  of  nonlinear  systems.  An  approach  has  been  developed  [6] 
that  allows  an  input  force  pattern  to  be  derived  that  will  result  in  a  non¬ 
linear  response  in  the  linear  mode  shape  of  interest.  This  force  pattern  is 
derived  using  an  optimisation  approach.  The  mode  of  interest  can  then  be 
identified  using  a  single  degree  of  freedom  nonlinear  identification  method. 
In  this  work  the  Restoring  Force  method  is  used  to  examine  the  nonlinear 
response  of  a  particular  linear  mode  and  an  application  of  this  approach 
to  a  two  degree  of  freedom  experimental  system  is  presented. 


THEORY 


The  theoretical  approach  is  demonstrated  for  the  two  degree  of  freedom 
system  with  spring  grounded  nonlinearity  shown  in  figure  1.  The  equations 
of  motion  for  this  system  in  physical  space  are: 


m  0 
0  m 


+ 


+ 


(1  +  a)c 

—ac 

'±1 1 

—ac 

(1  +  d)c 

\ 

X2  J 

(l  -}-  Q^k 

—ak 

fill 

—ak 

(1  +  a)k 

l^x\ 

0 


where  is  the  cubic  stifEhess  coefficient  and  a  is  a  constant  that  allows  the 
frequency  spacing  of  the  natural  frequencies  to  be  varied.  These  equations 
can  be  transformed  to  linear  modal  space  using  the  transformation: 


{i}  =  [ij>]{u}  (2) 

where  [(j>]  is  the  modal  matrix  of  the  underlying  linear  system  and  the 
vector  {u}  defines  the  modal  displacements.  For  this  symmetrical  system 


1100 


the  modal  matrix  is 


1  1 
1  -1 


(3) 


The  equations  of  motion  transformed  to  linear  modal  space  using  the  nor¬ 
malised  modal  matrix  are: 


771  0  \  ill 

0  771  (112 


+ 


c  0 

0  (1  +  2g)c 

1^2/ 

■jb  0 

Ull 

0  (1  +  2a)k 

1^2] 

P{Ul-U2y/4:' 

-U2)V4 


where  {p}  is  the  modal  input  vector.  It  can  be  seen  from  the  above  equa¬ 
tions  that  the  cubic  nonlinearity  couples  the  modes  in  linear  mod^  space; 
in  fact  there  are  a  significant  number  of  terms  for  a  single  nonlinearity. 
The  proposed  method  aims  to  determine  the  force  pattern  that  will  reduce 
the  response  of  this  system  to  that  of  a  single  mode. 

It  was  shown  in  a  previous  paper  [3]  that  this  can  be  achieved  by  seeking  a 
force  vector  that  will  cause  motion  only  in  the  target  mode,  by  eliminating 
motion  in  the  coupled  mode.  In  practice,  physical  data  from  transducers 
are  available.  Any  subsequent  transformations  would  be  time  consuming. 
It  is  shown  below  that  causing  motion  in  one  mode  to  be  zero  is  equiva¬ 
lent  to  forcing  motion  in  a  linear  mode  shape,  mode  one  in  this  example. 
Consider  the  coordinate  transformation  {a;}  =  [<^]{ii}  or  more  explicitly 
for  the  two  degree  of  freedom  system  in  Figure  1: 


Thus 


1 

2 


‘l  -1 
1  1 


(5) 

(6) 


and  enforcing  the  first  mode  shape  (1, 1)  in  physical  space  should  give  a 
second  modal  displacement  of  zero. 


1 

2 


1  -1 
1  1 


1 

1 


(7) 


So  if  an  excitation  is  applied  which  causes  the  nonlinear  system  to  vibrate 
in  its  first  linear  mode  shape,  the  response  will  be  composed  only  of  ui  and 
the  influence  of  the  coupled  mode,  U2,  will  then  have  been  eliminated. 

The  method  must  therefore  derive  a  force  pattern  which  will  cause  the 
system  to  vibrate  in  one  of  its  linear  mode  shapes.  It  has  been  shown 


1101 


in  a  previous  paper  [3]  that  if  the  response  contains  harmonics  then  the 
force  pattern  must  also  contain  harmonics  in  order  to  control  the  harmonic 
content  of  the  response.  In  theory,  the  responses  will  be  an  infinite  series 
of  harmonics,  but  this  series  is  truncated  in  this  case  of  a  cubic  stiffness 
nonlinearity  to  include  only  the  fundamental  and  third  harmonic  terms. 

The  physical  input  forces  will  thus  be  of  the  form: 

fi(t)  =  Fii  cos(a;ea;t  4-  ^ii)  4-  F13  cos(a;ea;t  +  (^13)  (8) 

f2{t)  =  F21  C0s(a;ea;t  +  (j>2l)  4*  F23  COS{uJext  +  fe)  (9) 

where  is  the  excitation  frequency.  Parameters  for  these  force  patterns 
may  then  be  chosen  such  that  only  mode  one  is  excited. 


OPTIMISATION  APPROACH 

In  general,  no  a  priori  model  of  the  system  exists  so  an  optimisation  routine 
is  used  to  determine  the  force  pattern  parameters  required  to  maximise  the 
contribution  of  the  mode  of  interest.  The  objective  function,  the  quantity 
that  the  optimisation  routine  seeks  to  minimise,  must  be  representative 
of  the  deviation  of  the  response  from  the  target  linear  mode  shape.  The 
objective  function,  F,  that  was  chosen  in  this  case  was  based  on  the  vector 
norm  [4]  of  the  two  physical  responses,  Xi  and  X2,  and  is  shown  below: 


where  and  02  are  elements  of  the  mode  shape  vector  for  the  target 
mode.  This  summation  is  carried  out  over  one  cycle  of  the  fundamental 
response.  The  number  of  data  points  per  cycle  is  npts  and  Xki  the  kth 
response  at  the  itk  sample.  This  objective  function  allows  the  response 
to  contain  harmonics  and  can  be  extended  to  more  degrees  of  freedom  by 
choosing  a  reference  displacement  and  subtracting  further  displacements 
from  it.  The  Variable  Metric  optimisation  method  [5]  was  used  in  this 
work  as  it  has  been  found  to  produce  the  best  results  for  simulated  data. 
The  application  of  this  method  to  a  two  degree  of  freedom  system  such 
as  that  shown  in  figure  1  is  detailed  in  [6].  Optimised  force  patterns  are 
obtained  at  several  levels  of  input  amplitude.  These  force  patterns  are  then 
applied  and  the  Restoring  Force  method  is  used  to  curve  fit  the  resulting 
modal  displacement  and  velocity  time  histories  to  give  the  direct  linear  and 
nonlinear  coefficients  for  the  mode  of  interest. 


1102 


EXPERIMENTAL  IMPLEMENTATION 


The  simulated  application  of  this  method  assumed  that  certain  parame¬ 
ters  were  known.  In  order  to  carry  out  an  identification  of  an  experimental 
structure,  these  parameters  must  be  measured  or  calculated.  Some  pro¬ 
cessing  of  experimental  data  is  necessary  in  order  to  apply  the  Restoring 
Force  method.  The  restoring  force  of  a  system  can  be  expressed  for  a  single 
degree  of  freedom  system  as: 

h{x,  x)  =  f(t)  -  mx  (11) 

where  h{x,  x)  is  the  restoring  force  and  f{t)  the  input  force.  A  similar  ex¬ 
pression  applies  to  the  modal  restoring  force  for  an  isolated  mode.  Thus  the 
input  force,  acceleration,  velocity  and  displacement  must  be  calculated  at 
the  each  time  instant.  A  similar  expression  applies  to  the  modal  restoring 
force  for  an  isolated  mode.  In  the  experimentaJ  case  it  is  usual  to  measure 
acceleration  and  input  force;  the  remaining  two  states  must  therefore  be 
obtained  by  integration  of  the  acceleration  time  history.  Frequency  domain 
integration  [7]  was  used  for  this  purpose.  High  pass  filtering  was  used  to 
remove  any  low  frequency  noise  which  can  be  amplified  by  this  type  of 
integration.  Several  methods  have  been  suggested  for  estimating  modal 
mass,  but  in  this  study  a  method  developed  by  Worden  and  Tomlinson  [8] 
was  used.  An  estimate  for  the  modal  mass  is  obtained  and  then  an  error 
term  is  included  in  the  curve  fit  which  will  iteratively  yield  a  more  accu¬ 
rate  estimate  of  the  mass.  Generally  the  mass  value  will  converge  after  one 
iteration. 

The  objective  function  used  in  the  simulations  was  calculated  from  the 
displacement  time  histories.  In  the  experimental  case,  acceleration  was 
used  rather  than  displacement  as  it  was  considered  that  using  ’raw’  data 
would  be  quicker  and  give  less  opportunity  for  error.  In  the  simulated 
case,  the  system  parameters  were  known  a  priori  so  the  modal  matrix 
of  the  underlying  linear  system  could  be  calculated.  For  most  types  of 
nonlinearity  the  response  of  the  system  at  low  input  force  levels  will  be 
dominated  by  linear  terms.  Normal  mode  tuning  [1]  was  therefore  applied 
at  low  force  levels  to  yield  an  approximation  to  the  modal  matrix  of  the 
underlying  linear  system. 

A  quality  indicator  to  give  some  idea  of  the  effectiveness  of  the  optimisa¬ 
tion  performed  would  be  advantageous.  Results  corrupted  by  background 
noise,  for  example,  could  then  be  discarded.  A  perfect  optimisation  will 
occur  when  the  ratio  of  measured  accelerations  exactly  matches  the  mode 
shape  ratio  specified  for  the  mode  of  interest.  Thus  a  least  squares  fit  of 
the  sampled  accelerations  was  carried  out  over  a  cycle  of  the  fundamental 
frequency  and  the  percentage  error  of  the  measured  mode  shape  to  the 


1103 


required  mode  shape  was  calculated.  This  percentage  error  will  indicate 
whether  the  optimisation  has  been  successful. 

To  assess  the  accuracy  of  the  parameters  estimated  using  this  method, 
an  identification  was  carried  out  using  a  conventional  Restoring  Force  ap¬ 
proach  in  physical  space.  A  band  limited  random  excitation  was  used,  and 
the  physical  data  processed  and  curve  fitted.  The  physical  parameters  were 
then  transformed  to  modal  space.  The  direct  linear  and  nonlinear  param¬ 
eters  for  modes  one  and  two  are  shown  in  table  1.  It  should  be  noted  that 
although  this  conventional  Restoring  Force  approach  is  possible  for  this 
two  degree  of  freedom  system,  it  will  not  generally  be  possible  since  the 
number  of  terms  in  the  curve  fit  increases  dramatically  when  different  t3q)es 
of  nonlinearity  and  more  degrees  of  freedom  are  included.  It  is  carried  out 
in  this  case  as  a  means  of  validating  the  proposed  method. 


EXPERIMENTAL  SETUP 

The  rig  constructed  consisted  of  two  masses  on  thin  legs  connected  in  series 
by  a  linear  spring,  each  mass  being  driven  by  a  shaker.  A  cubic  nonlin¬ 
earity  was  introduced  between  the  first  mass  and  ground  using  a  clamped- 
clamped  beam  attached  at  the  centre  which  will  yield  a  cubic  stiffness  for 
large  deflections  [9],  A  schematic  diagram  of  the  rig  is  shown  in  figure  2. 
The  force  input  by  each  shaker  was  measured  using  a  force  gauge  and  the 
acceleration  of  each  mass  was  measured  using  an  accelerometer  in  the  po¬ 
sitions  also  shown  in  figure  2,  Acceleration  and  force  data  were  acquired 
using  a  multiple  channel  acquisition  system,  the  optimisation  routine  was 
carried  out  on  line. 


RESULTS 


Normal  mode  tuning  of  the  rig  gave  natural  frequencies  of  20.67  Hz  and 
24.27  Hz  and  a  modal  matrix  of: 


3.87  5.03 
5.52  -3.27 


(12) 


The  excitation  frequency  was  chosen  to  be  slightly  lower  than  the  natural 
frequency  of  the  mode  of  interest  in  order  to  avoid  the  problems  associated 
with  force  drop  out  which  are  worst  at  the  natural  frequency.  For  each 
mode  optimisation  was  performed  at  three  input  force  levels,  the  highest 
level  was  as  high  as  possible  so  as  to  excite  the  nonlinearity  strongly.  The 
optimisation  routine  was  carried  out  using  the  voltage  input  into  the  signal 
generator  as  the  variable.  The  force  input  into  the  structure  was  measured 


1104 


for  use  in  the  Restoring  Force  identification  but  was  not  used  in  the  opti¬ 
misation  as  it  is  not  directly  controllable.  The  details  of  the  optimisation 
for  each  force  level  are  presented  in  table  2.  The  optimised  forces  and  re¬ 
sulting  accelerations  for  mode  one  are  shown  in  figure  3  and  figure  4.  The 
acceleration  data  for  the  optimised  force  patterns  were  then  integrated  and 
the  modal  restoring  force  for  the  mode  of  interest  calculated.  The  initial 
estimate  of  modal  mass  for  the  calculation  of  the  modal  restoring  force  was 
tahen  from  a  previous  paper  [10]  in  which  the  rig  was  identified  using  a 
using  a  physical  parameter  identification  method.  The  mass  was  estimated 
in  this  paper  to  be  2.62  kg,  this  physical  mass  will  then  be  equal  to  the 
modal  mass  since  the  modal  matrix  was  normalised  to  be  orthonormal. 
The  restoring  force  data  was  then  transformed  to  modal  space.  The  modal 
restoring  force  surface  obtained  using  optimised  force  inputs  for  mode  one 
is  shown  in  figure  5  and  a  stiffness  section  through  this  surface  is  shown  in 
figure  6.  The  restoring  force  time  histories  were  then  curve  fitted  against 
modal  velocity  and  displacement. 

The  estimated  parameters  for  mode  one  axe  shown  in  table  3.  It  can  be 
seen  that  they  do  not  compare  very  well  with  those  estimates  obtained 
using  the  band  limited  random  excitation.  It  was  suspected  that  at  lower 
excitation  levels  the  estimates  were  being  distorted  by  linear  dependence 
[11].  Linear  dependence  is  a  problem  which  occurs  when  curve  fitting  a 
harmonic  response  from  a  linear  system;  the  equations  of  motion  may  be 
identically  satisfied  by  mass  and  stiffness  terms  modified  by  an  arbitrary 
constant.  This  condition  is  avoided  by  the  harmonic  terms  introduced  into 
the  response  by  nonlinearity.  The  curve  fit  was  thus  repeated  using  only  the 
data  obtained  from  the  highest  level  of  excitation;  the  estimates  obtained 
are  shown  in  table  4.  It  can  be  seen  that  the  linear  parameter  estimates 
now  agree  well  with  the  band  limited  random  results.  The  estimates  for 
the  cubic  stiffness  coefficient  do  not  appear  to  agree  so  well.  The  standard 
deviation  on  the  cubic  stiffness  derived  from  the  band  limited  random 
excitation  is  approximately  a  third  of  the  value  of  the  parameter  itself.  The 
uncertainty  on  this  parameter  occurs  because  the  nonlinearity  is  not  very 
strongly  excited  by  this  type  of  excitation.  A  stiffness  section  through  the 
restoring  force  surface,  figure  7,  shows  little  evidence  of  a  cubic  stiffness 
component.  If  a  higher  level  of  excitation  were  possible  then  a  better 
estimate  may  be  achieved. 

The  identification  w^  repeated  for  mode  two.  The  restoring  force  surface 
obtained  and  a  stiffness  section  through  it  are  shown  in  figure  8  and  figure  9. 
It  can  be  seen  from  the  stiffness  section  that  the  nonlinearity  is  not  very 
strongly  excited.  The  estimated  direct  modal  parameters  are  shown  in 
table  5.  It  can  be  seen  that  these  results  agree  quite  well  with  those 
obtained  using  band  limited  random  excitation.  It  is  considered  that  the 
discrepancy  between  the  two  sets  of  results,  in  particular  the  mass  and 


1105 


stiifiiess  estimates,  is  again  due  to  linear  dependence. 


CONCLUSIONS 

An  extension  of  the  force  appropriation  method  has  been  proposed  for 
nonlinear  systems.  In  this  method,  an  optimisation  routine  is  used  to 
determine  the  force  patterns  which  will  excite  a  single  mode  nonlinear 
response  .  The  direct  linear  and  nonlinear  modal  parameters  can  then  be 
estimated  from  a  curve  fit  of  the  modal  restoring  force  surface.  The  method 
was  applied  to  an  experimental  two  degree  of  freedom  system  whose  modes 
were  coupled  in  linear  modal  space  by  a  spring  grounded  nonlinearity. 
A  conventional  restoring  force  identification  was  performed  using  a  band 
limited  random  signal  for  comparison.  The  parameters  estimated  from  the 
single  mode  responses  were  found  to  agree  quite  well  with  those  from  the 
band  limited  random  tests. 


REFERENCES 


1.  Holmes  P.,  Advanced  Applications  of  Normal  Mode  Testing,  PhD 
Thesis,  University  of  Manchester  1996. 

2.  Hadid  M.A.  and  Wright  J.R.,  Application  of  Force  State  Mapping 
to  the  Identification  of  Nonlinear  Systems  Mechanical  Systems  and 
Signal  Processing,  1990,  4(6),  463-482 

3.  Atkins  P.A.,  Wright  J.R.,  Worden  K.,  Manson  G.M.  and  Tomlinson 
G.R.,  Dimensional  Reduction  for  Multi  Degree  of  Freedom  Nonlinear 
Systems,  International  Conference  on  Identification  in  Engineering 
Systems  1996,  712-721 

4.  Kreyszig  E.,  Advanced  Engineering  Mathematics,  Wiley,  6th  Edition 

5.  Press  W.H.,  Teukolsky  S.A.,  Vettering  W.T.  and  Flannery  B.P.,  Nu¬ 
merical  Recipes  in  Fortran  Cambridge  University  Press,  6th  Edition 

6.  Atkins  P.A.  and  Wright  J.R.,  An  Extension  of  Force  Appropriation 
for  Nonlinear  Systems  Noise  and  Vibration  Engineering,  Proceedings 
of  ISMA21(2),  915-926,  1996 

7.  Worden  K.,  Data  Processing  and  Experiment  Design  for  the  Restor¬ 
ing  Force  Method,  Part  I:  Integration  and  Differentiation  of  Measured 
Time  Data,  Mechanical  Systems  and  Signal  Processing,  4(4)  295-319, 
1990 


1106 


8.  Ajjan  Al-Hadid  M.,  Identification  of  Nonlinear  Dynamic  Systems  us¬ 
ing  the  Force  State  Mapping  Technique,  PhD  Thesis,  Queen  Mary 
College,  University  of  London,  1989 

9.  Storer  D.M.,  Dynamic  Analysis  of  Nonlinear  Structures  Using  Higher 
Order  Frequency  Response  Function,  PhD  Thesis,  University  of  Manch¬ 
ester,  1991 

10.  Atkins  P.  and  Worden  K.,  Identification  of  a  Multi  Degree  of  Freedom 
Nonlinear  System,  Proc.  of  IMAC  XV,  1997 


ACKNOWLEDGEMENTS 

This  work  was  supported  by  E.P.S.R.C.  under  research  grant  number 
GR/J48238  at  the  University  of  Manchester 


Modal  parameter 

Mode  one 

Mode  two 

k  (N/m) 
c  (Nm/s) 

/3(N/m^) 
m  (kg) 

4.87  X  10^ 
10.11 

3.83  X  10® 
2.60 

6.49  X  10^ 
9.49 

8.90  X  10* 
3.06 

Table  1:  Direct  modal  parameters  estimated  from  curve  fit  of  band  limited 
random  data 


Low  forcing 

Medium  forcing 

High  forcing 

Fii  (Volts) 

1.0 

2.0 

3.0 

F21  (Volts) 

1.0 

2.0 

3.0 

u)^  (Hz) 

20.0 

20.0 

20.0 

initial  mode  shape  ratio 

0.72 

0.68 

0.65 

final  mode  shape  ratio 

0.70 

0.70 

0.70 

target  mode  shape  ratio 

0.70 

0.70 

0.70 

percentage  error 

0.14 

0.03 

0.71 

Table  ,2:  Details  of  optimisation  for  mode  one 


Table  3:  Direct  modal  parameters  estimated  from  optimised  responses 


Model  parameter  Estimated  parameter 
k  (N/m)  4.57  X  10^ 

c  (Nm/s)  8.62 

/?(N/m3)  6.81  X  10® 

m  (kg) _ _ 2^75 _ 

Table  4:  Direct  modal  parameters  estimated  using  high  force  level  only 


Model  parameter 

Estimated  parameter 

k  (N/m) 
c  (Nm/s) 

m  (kg) 

5.37  X  10^ 

10.16 

2.45  X  10® 

2.29 

Table  5:  Direct  modal  parameters  estimated  for  mode  two 


Figure  1:  Two  degree  of  freedom  system 


1108 


0  s  0.40008 


Figure  3:  Optimised  forces  for  mode  one  at  a  high  force  level 


Figure  4:  Accelerations  responses  to  optimised  forces 


Figure  5:  Modal  restoring  force  surface  for  mode  one 


Force  (N) 


Figure  7:  Stiffness  section  through  modal  restoring  force  derived  from  ran¬ 
dom  excitation  for  mode  one 


1111 


Figure  8:  Modal  restoring  force  surface  for  mode  two 


Figure  9:  Stiffness  section  through  modal  restoring  force  for  mode  two 


1112 


POWER  FLOW  TECHNIQUES  II 


THE  OPTIMAL  DESIGN  OF  NEAR-PERIODIC  STRUCTURES  TO 
MINIMISE  NOISE  AND  VIBRATION  TRANSMISSION 


R.S.  Langley,  N.S.  Bardell,  and  P.M.  Loasby 
Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 
Southampton  SO  17  IBJ,  UK 

1.  INTRODUCTION 

An  engineering  structure  is  said  to  be  of  "periodic"  construction  if  a  basic 
structural  unit  is  repeated  in  a  regular  pattern  .  A  beam  which  rests  on 
regularly  spaced  supports  is  one  example  of  a  one-dimensional  periodic 
structure,  while  an  orthogonally  stiffened  cylinder  is  an  example  of  a  two- 
dimensional  periodic  structure.  It  has  long  been  known  that  perfectly  periodic 
structures  have  very  distinctive  vibration  properties,  in  the  sense  that  "pass 
bands"  and  "stop  bands"  arise:  these  are  frequency  bands  over  which  elastic 
wave  motion  respectively  can  and  cannot  propagate  through  the  structure  [1,2]. 
If  the  excitation  frequency  lies  within  a  stop  band  then  the  structural  response 
tends  to  be  localised  to  the  immediate  vicinity  of  the  excitation  source. 
Conversely,  if  the  excitation  frequency  lies  within  a  pass  band  then  strong 
vibration  transmission  can  occur,  and  it  is  generally  the  case  that  the  resonant 
frequencies  of  the  structure  lie  within  the  pass  bands. 

Much  recent  work  has  been  performed  concerning  the  effect  of  random 
disorder  on  a  nominally  periodic  structure  (see  for  example  [3-5]).  It  has  been 
found  that  disorder  can  lead  to  localisation  of  the  response  even  for  excitation 
which  lies  within  a  pass  band,  and  this  reduces  the  propensity  of  the  structure 
to  transmit  vibration.  This  raises  the  possibility  of  designing  disorder  into  a 
structure  in  order  to  reduce  vibration  transmission,  and  this  possibility  was 
briefly  investigated  in  reference  [6]  for  a  one-dimensional  periodic  waveguide 
which  was  embedded  in  an  otherwise  infinite  homogeneous  system.  The 
present  work  extends  the  work  reported  in  reference  [6]  to  the  case  of  a  finite 
near-periodic  beam  system,  which  more  closely  resembles  the  type  of 
optimisation  problem  which  is.  likely  to  occur  in  engineering  practice.  The 
beam  is  taken  to  have  N  bays,  and  the  design  parameters  are  taken  to  be  the 
individual  bay  lengths.  Both  single  frequency  and  band-limited  excitation  are 
considered,  and  two  objective  functions  are  investigated:  (i)  the  response  in  a 
bay  which  is  distant  from  the  applied  loading  (minimisation  of  vibration 
transmission),  and  (ii)  the  maximum  response  in  the  structure  (minimisation 
of  maximum  stress  levels).  In  each  case  the  optimal  configuration  is  found  by 
employing  a  quasi-Newton  algorithm,  and  the  physical  features  of  the  resulting 
design  are  discussed  in  order  to  suggest  general  design  guidelines. 


1113 


2.  ANALYTICAL  MODEL  OF  THE  NEAR-PERIODIC  BEAM 


2.1  Calculation  of  the  Forced  Response 

A  schematic  of  an  N  bay  near-periodic  beam  structure  is  shown  in  Figure  1. 
The  structure  is  subjected  to  dynamic  loading,  and  the  aim  of  the  present  work 
is  to  find  the  optimal  design  which  will  minimise  a  prescribed  measure  of  the 
vibration  response.  No  matter  what  type  of  optimisation  algorithm  is 
employed,  this  type  of  study  requires  repeated  computation  of  the  system 
dynamic  response  as  the  design  parameters  are  varied,  and  it  is  therefore 
important  to  employ  an  efficient  analysis  procedure.  In  the  present  work  the 
h-p  version  of  the  finite  element  method  (FEM)  is  employed:  with  this 
approach  the  structure  is  modelled  as  an  assembly  of  elements  which  have 
both  nodal  and  internal  degrees  of  freedom.  Each  element  has  two  nodes  and 
the  nodal  degrees  of  freedom  consist  of  the  beam  displacement  and  slope;  the 
internal  degrees  of  freedom  are  generalised  coordinates  which  are  associated 
with  a  hierarchy  of  shape  functions  which  contribute  only  to  the  internal 
displacement  field  of  the  element.  The  internal  shape  functions  used  here  are 
the  K-orthogonal  Legendre  polynomials  of  order  four  onwards  -  full  details  of 
the  present  modelling  approach  are  given  in  reference  [7]. 

For  harmonic  excitation  of  frequency  co  the  equations  of  motion  of  the 
complete  beam  structure  can  be  written  in  the  form 

where  M  and  K  are  the  global  mass  and  stiffness  matrices  (assembled  from  the 
individual  element  matrices  taking  into  account  the  presence  of  any  mass  or 
spring  elements  and  allowing  for  constraints),  q  contains  the  system 
generalized  coordinates,  F  is  the  generalized  force  vector,  and  r|  is  the  loss 
factor,  which  in  the  present  study  is  taken  to  be  uniform  throughout  the 
structure. 

Equation  (1)  can  readily  be  solved  to  yield  the  system  response  q.  In  the 
present  work  it  is  convenient  to  use  the  time  averaged  kinetic  and  strain 
energies  of  each  of  the  N  bays  as  a  measure  of  the  response  -  for  the  nth  bay 
these  quantities  can  be  written  as  and  say,  where 

r,=(£0V4)9;>„?„.  (2.3) 

Here  and  K„  are  the  mass  and  stiffness  matrices  of  the  nth  bay,  and  q„  is 
the  vector  of  generalized  coordinates  for  this  bay. 

Many  of  the  physical  features  of  the  forced  response  of  a  near-periodic 
structure  can  be  explained  in  terms  of  the  free  vibration  behaviour  of  the 


1114 


associated  perfectly  periodic  structure.  The  following  section  outlines  how  the 
present  finite  element  modelling  approach  can  be  used  to  study  the  pass  bands 
and  stop  bands  exhibited  by  a  perfect  periodic  structure. 

2.2  Periodic  Structure  Analysis 

The  finite  element  method  described  in  section  2.1  can  be  applied  to  a  single 
bay  of  a  perfectly  periodic  structure  to  yield  an  equation  of  motion  in  the  form 

Dq=F,  D=-coW--(l+/r|)ii:,  (^’5) 


where  the  matrix  D  is  referred  to  as  the  dynamic  stiffness  matrix.  In  order  to 
study  wave  motion  through  the  periodic  system  it  is  convenient  to  partition  D, 
q  and  F  as  follows 


r 

Du 

ft: 

D= 

D„ 

D,r 

.  F= 

0 

Dr, 

F„ 

V  V 

(6-8) 


where  L  relates  to  the  coordinates  at  the  left  most  node,  R  relates  to  those  at 
the  right  most  node,  and  /  relates  to  the  remaining  "internal"  coordinates. 
Equations  (4-8)  can  be  used  to  derive  the  following  transfer  matrix  relation 
between  the  displacements  and  forces  at  the  left  and  right  hand  nodes 


V  / 


V  / 


(9,10) 

Equation  (9)  can  now  be  used  to  analyze  wave  motion  through  the  periodic 
system:  such  motion  is  governed  by  Bloch’s  Theorem,  which  states  that 
i^L  F i)=exp{-ie-b){qii  -F^)  where  8  and  5  are  known  respectively  as  the  phase 
and  attenuation  constants.  A  pass  band  is  defined  as  a  frequency  band  over 
which  6=0,  so  that  wave  motion  can  propagate  down  the  structure  without 
attenuation.  It  follows  from  equation  (9)  that 


(T-Ie 


-ie-6' 


(  \ 

0 

F, 

io 

V 

V  y 

(11) 


SO  that  8  and  6  can  be  computed  from  the  eigenvalues  of  T,  thus  enabling  the 
pass  bands  and  stop  bands  to  be  identified. 


1115 


2.3  Optimisation  Procedure 

Equations  (l)-(3)  enable  the  forced  response  of  the  system  to  be  calculated  for 
any  prescribed  set  of  system  properties.  The  aim  of  the  present  analysis  is  to 
compute  the  optimal  set  of  system  properties  for  a  prescribed  design  objective, 
and  in  order  to  achieve  this  equations  (l)-(3)  are  evaluated  repeatedly  as  part 
of  an  optimisation  algorithm.  As  an  example,  it  might  be  required  to  minimise 
the  kinetic  energy  of  bay  N  by  changing  the  various  bay  lengths.  In  this  case 
equations  (l)-(3)  provide  the  route  via  which  the  objective  function  (the  kinetic 
energy  in  bay  N)  is  related  to  the  design  parameters  (the  bay  lengths),  and  the 
optimisation  algorithm  must  adjust  the  design  parameters  so  as  to  minimise  the 
objective  function.  The  optimisation  process  has  been  performed  here  by  using 
the  NAg  library  routine  E04JAF  [8],  which  employs  a  quasi-Newton  algorithm. 
This  type  of  algorithm  locates  a  minimum  in  the  objective  function,  although 
there  is  no  indication  whether  this  minimum  is  the  global  minimum  or  a  less 
optimal  local  minimum.  The  probability  of  locating  the  global  minimum  can 
be  increased  significantly  by  repeated  application  of  the  NAg  routine  using 
random  starts,  i.e.  random  initial  values  of  the  design  parameters.  Numerical 
investigations  have  led  to  the  use  of  30  random  starts  in  the  present  work. 

3.  NUMERICAL  RESULTS 

3.1  The  System  Considered 

The  foregoing  analysis  has  been  applied  to  a  beam  of  flexural  rigidity  El,  mass 
per  unit  length  m,  and  loss  factor  r|=0.015,  which  rests  on  A+1  simple 
supports,  thus  giving  an  A’-bay  near-periodic  system.  The  design  parameters 
are  taken  to  be  the  bay  lengths  (i.e.  the  separation  of  the  simple  supports),  and 
the  design  is  constrained  so  that  the  length  of  any  bay  lies  within  the  range 
0.9L^,<1.1L,  where  L,  is  a  reference  length.  A  non-dimensional  frequency 
Q  is  introduced  such  that  0.=(oL,N{m/Er),  and  the  non-dimensional  kinetic  and 
strain  energies  of  a  bay  are  defined  as  T,'-T,^(EI/L^^\F\^)  and 
U,'~U„iEI/L,^\F\'^)  where  F  is  the  applied  point  load.  As  discussed  in  the 
following  subsections,  two  objective  functions  are  considered  corresponding 
to  minimum  vibration  transmission  and  minimum  overall  response.  In  all 
cases  the  excitation  consists  of  a  point  load  applied  to  the  first  bay  and  the 
response  is  averaged  over  1 1  equally  spaced  point  load  locations  within  the 
bay.  For  reference,  the  propagation  constants  for  a  periodic  system  in  which 
all  the  bay  lengths  are  equal  to  L,  are  shown  in  Figure  2  -  the  present  study  is 
focused  on  excitation  frequencies  which  lie  in  the  range  23<f2<61,  which 
covers  the  second  stop  band  and  the  second  pass  band  of  the  periodic  system. 

3.2  Design  for  Minimum  Vibration  Transmission 

In  this  case  the  objective  function  is  taken  to  be  the  kinetic  energy  in  bay  N, 
so  that  the  aim  is  to  minimise  the  vibration  transmitted  along  the  structure. 
Three  types  of  loading  are  considered;  (i)  single  frequency  loading  with  Q=50, 


1116 


which  lies  within  the  second  pass  band  of  the  ordered  structure;  (ii)  band- 
limited  loading  with  40<Q<60,  which  covers  the  whole  of  the  second  pass 
band;  (iii)  band-limited  loading  with  23<Q<61,  which  covers  the  whole  of  the 
second  stop  band  and  the  second  pass  band. 

Results  for  the  optimal  design  under  single  frequency  loading  are  shown  in 
Table  1;  in  all  cases  it  was  found  that  the  bay  lengths  were  placed  against 
either  the  upper  bound  (U=1.1L,)  or  the  lower  bound  (L=0.9L,),  and  significant 
reductions  in  the  energy  level  of  bay  N  were  achieved.  In  this  regard  it  should 
be  noted  that  the  dB  reduction  quoted  on  Table  1  is  defined  as  -101og(r;v/^;^r) 
where  is  the  kinetic  energy  in  the  final  bay  of  the  ordered  system.  The 
optimal  designs  shown  in  Table  1  all  tend  to  consist  of  a  bi-periodic  structure 
in  which  the  basic  unit  consists  of  two  bays  in  the  configuration  LU.  The  pass 
bands  and  stop  bands  for  this  configuration  are  shown  in  Figure  3,  and  further, 
for  the  optimal  12  bay  system  is  shown  in  Figure  4  over  the  frequency 
range  0<Q<250.  By  comparing  Figures  3  and  4  it  is  clear  why  the  selected 
design  is  optimal  -  the  new  bi-periodic  system  has  a  stop  band  centred  on  the 
specified  excitation  frequency  Q=50.  It  can  be  seen  from  Figure  4  that  the 
improvement  in  the  response  at  the  specified  frequency  0=50  is  accompanied 
by  a  worsening  of  the  response  at  some  other  frequencies. 

Results  for  the  optimal  design  under  band-limited  excitation  over  the  range 
40<Q<60  are  shown  in  Table  2.  In  some  cases  two  results  are  shown  for  the 
optimised  "Final  Energy":  in  such  cases  the  first  result  has  been  obtained  by 
forcing  each  bay  length  onto  either  the  upper  (U)  or  lower  (L)  bound,  while 
the  second  result  has  been  obtained  by  using  the  NAg  optimisation  routine. 
If  only  one  result  is  shown  then  the  two  methods  yield  the  same  optimal 
design.  The  "bound"  result  is  easily  obtained  by  computing  the  response  under 
each  possible  combination  of  U  and  L  bay  lengths  -  this  requires  2^  response 
calculations,  which  normally  takes  much  less  CPU  time  than  the  NAg 
optimisation  routine.  It  is  clear  from  Table  2  that  the  additional  improvement 
in  the  response  yielded  by  the  full  optimisation  routine  is  minimal  for  this 
case.  The  response  curve  for  the  12-bay  system  is  shown  in  Figure  5,  where 
it  is  clear  that  a  significantly  reduced  response  is  achieved  over  the  specified 
frequency  range;  as  would  be  expected  an  increase  in  the  response  can  occur 
at  other  frequencies.  It  is  interesting  to  note  that  most  of  the  optimal  designs 
shown  in  Table  2  lack  symmetry  -  however,  it  follows  from  the  principle  of 
reciprocity  that  a  design  which  minimises  vibration  transmission  from  left  to 
right  will  also  minimise  transmission  from  right  to  left.  It  should  therefore  be 
possible  to  "reverse"  the  designs  without  changing  the  transmitted  vibration 
levels.  This  hypothesis  is  tested  in  Figure  6  for  a  12  bay  structure  -  the  figure 
shows  the  energy  distribution  for  the  optimal  design  UUULUULLLLLU  and 
for  the  reversed  design  ULLLLLUULUUU.  Although  the  detailed  distribution 
of  energy  varies  between  the  two  designs,  the  energy  levels  achieved  in  bay 


1117 


12  are  identical,  as  expected. 


Results  for  the  optimal  design  under  wide-band  excitation  23<Q<61  are  shown 
in  Table  3,  and  the  response  curve  for  the  12-bay  optimised  system  is  shown 
in  Figure  7.  The  form  of  optimal  design  achieved  is  similar  to  that  obtained 
for  the  narrower  excitation  band  40<Q<60,  although  there  are  detailed 
differences  between  the  two  sets  of  results.  In  each  case  there  is  a  tendency 
for  a  group  of  lower  bound  bays  (L)  to  occur  in  the  mid  region  of  the 
structure,  and  a  group  of  upper  bound  bays  (U)  to  occur  at  either  end.  This 
creates  an  "impedance  mismatch"  between  the  two  sets  of  bays,  which 
promotes  wave  reflection  and  thus  reduces  vibration  transmission  along  the 
structure.  By  comparing  Tables  1-3,  it  is  clear  that  the  achievable  reduction 
in  vibration  transmission  reduces  as  the  bandwidth  of  the  excitation  is 
increased. 

3.3  Design  for  Minimum  ''Maximum”  Strain  Energy 

In  this  case  the  strain  energy  U„  of  each  bay  is  computed  and  the  objective 
function  is  taken  to  be  the  maximum  value  of  U„.  As  a  design  objective,  this 
procedure  can  be  likened  to  minimising  the  maximum  stress  in  the  structure. 
As  in  the  previous  section  the  three  frequency  ranges  Q=50,  40<n<60,  and 
23<Q<61  are  considered,  and  the  present  study  is  limited  to  systems  having 
9,10,  11,  or  12  bays;  the  optimal  designs  achieved  are  shown  in  Table  4. 

Considering  the  single  frequency  results  (Q.=50)  shown  in  Table  4,  it  is  clear 
that  a  large  dB  reduction  is  achieved  only  for  those  systems  which  have  an 
even  number  of  bays;  furthermore,  the  optimal  energy  obtained  has  the  same 
value  (0.0297)  in  all  cases.  This  can  be  explained  by  noting  that  for  an  odd 
number  of  bays  the  frequency  Q=50  lies  near  to  an  anti-resonance  of  the 
ordered  structure,  whereas  a  resonance  is  excited  for  an  even  number  of  bays  - 
this  feature  is  illustrated  in  Figure  8  for  the  12  bay  structure.  The  repeated 
occurrence  of  the  optimal  energy  0.0297  arises  from  the  fact  that  the  initial  bay 
pattern  ULLLUUU  occurs  in  all  four  designs  -  it  has  been  found  that  this 
pattern  causes  a  vibration  reduction  of  over  20dB  from  bay  1  to  bay  8,  so  that 
the  response  in  bay  1  (the  maximum  response)  is  insensitive  to  the  nature  of 
structure  from  bay  8  onwards. 

The  optimal  "bounded"  designs  arising  for  band-limited  excitation  either  tend 
to  be  of  the  "UL"  bi-periodic  type  or  else  nearly  all  the  bays  are  assigned  the 
same  length.  However  it  should  be  noted  that  in  all  cases  the  design  produced 
by  the  NAg  optimisation  routine  offers  an  improvement  over  the  "bounded" 
design,  particularly  for  the  wide-band  case  (23<n<61).  It  is  clear  from  Table 
4  that  the  achieved  reduction  in  strain  energy  reduces  as  the  bandwidth  of  the 
excitation  is  increased. 


1118 


4.  CONCLUSIONS 


The  present  work  has  considered  the  optimal  design  of  a  near-periodic  beam 
system  to  minimise  vibration  transmission  and  also  maximum  stress  levels. 
With  regard  to  vibration  transmission  it  has  been  found  that  very  significant 
reductions  in  transmission  are  achievable  with  relatively  minor  design  changes. 
The  optimum  design  normally  involves  placing  the  design  parameters  (the  bay 
lengths)  on  the  permissible  bounds,  and  this  means  that  a  simple  design  search 
routine  can  be  used  in  preference  to  a  full  optimisation  algorithm.  With  regard 
to  minimum  stress  levels,  it  has  been  found  that  the  optimal  design  for  wide¬ 
band  excitation  is  not  normally  a  "bounded"  design,  and  thus  use  of  a  full 
optimisation  algorithm  is  preferable  for  this  case.  For  both  vibration 
transmission  and  maximum  stress  levels,  the  benefits  obtained  from  an  optimal 
design  decrease  with  increasing  excitation  bandwidth,  but  nonetheless  very 
significant  reductions  can  be  obtained  for  wide-band  excitation. 

REFERENCES 

1.  S.S.  MESTER  and  H.  BENAROYA  1995  Shock  and  Vibration  2,  69- 
95.  Periodic  and  near-periodic  structures. 

2.  D.J.  MEAD  1996  Journal  of  Sound  and  Vibration  190,  495-524.  Wave 
propagation  in  continuous  periodic  structures:  research  contributions 
from  Southampton  1964-1995. 

3.  C.H.  HODGES  1982  Journal  of  Sound  and  Vibration  82,  411-424. 
Confinement  of  vibration  by  structural  irregularity. 

4.  D.  BOUZIT  and  C.  PIERRE  1992  Journal  of  Vibration  and  Acoustics 
114,  521-530.  Vibration  confinement  phenomena  in  disordered,  mono- 
coupled,  multi-span  beams. 

5.  R.S.  LANGLEY  1996  Journal  of  Sound  and  Vibration  189,  421-441. 
The  statistics  of  wave  transmission  through  disordered  periodic 
waveguides. 

6.  R.S.  LANGLEY  1995  Journal  of  Sound  and  Vibration  188,  717-743. 
Wave  transmission  through  one-dimensional  near  periodic  structures: 
optimum  and  random  disorder. 

7.  N.S.  BARDELL,  R.S.  LANGLEY,  J.M.  DUNSDON  and  T.  KLEIN 
1996  Journal  of  Sound  and  Vibration  197,  427-446.  The  effect  of 
period  asymmetry  on  wave  propagation  in  periodic  beams. 

8.  ANON  1986  The  NAg  Fortran  Workshop  Library  Handbook  -  Release 
1.  Oxford:  NAg  Ltd. 


1119 


TABLE  1 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  D.=50. 


Original  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 
Final  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


No.  of 

Optimal  Pattern 

Original 

Final 

Reduction 

Bays,  N 

Energy 

Energy 

(dB) 

4 

UULU 

0.276E  1 

0.804E-3 

35.348 

ULULU 

0.609E-1 

0.179E-3 

■S 

UULULULU 

0.674E  0 

0.613E-5 

ULULULULU 

0.564E-1 

0.135E-5 

46.216 

1— 

10 

UULULULULU 

0.424E  0 

0.532E-6  ^ 

11 

ulululululu 

0.535E-1 

0.n7E-6 

56.604 

12 

UULULULULULU 

0.289E  0 

0.461E-7 

67.966 

13 

ULULULULULULU 

0.502E-1 

O.lOlE-7 

66.950 

16 

UULULULULULULULU 

0.154E0 

0.346E-9 
_ _ — 

86.484 

17 

ululululululululu 

0.43  lE-1 

0.761E-10 

87.529  1 

1120 


TABLE  2 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  40<Q<60. 

Original  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


No.  of 

Optimal  pattern 

Original  | 

Final 

Reduction 

Bays,  N 

Energy 

Energy 

(dB) 

■■ 

ULLU 

0.670E  0 

0.103E-1 

18.112 

5 

ULLLU 

0.63  IE  0 

0.735E-2 

19.338 

0.711E-2 

19.482 

6 

UULLLU 

0.221E-2 

22.407 

7 

ULLUULU 

0.463E  0 

0.171E-2 

24.335 

8 

UULLLLLU 

0.430E  0 

0.966E-3 

26.487 

0.914E-3 

26.725 

9 

UUULLLLLU 

0.444E  0 

0.341E-3 

31.142 

10 

UUUULLLLLU 

0.449E  0 

0.192E-3 

33.681 

0.189E-3 

33.758 

11 

ULLUUUULLLU 

0.291E0 

0.821E-4 

35.504 

12 

UUULUULLLLLU 

0.201E0 

0.352E-4 

37.558 

13 

1  ULUUUUULLLLLU 

0.199E0 

0.153E-4 

41.148 

1121 


TABLE  3 

Optimal  design  of  1-D  beam  structure,  to  minimise  energy  transmission,  23<Q<6L 

Original  Energy;  Non-dimensional  kinetic  energy  in  bay  N  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  kinetic  energy  in  bay  N  of  the  optimised  structure. 


6 

LLLLUU 

7 

LLLULUU 

8 

LLLLLUUU 

9 

ULLLLLLUU 

0.494E  0 


0.183E0 

0.175E0 


0.648E-2 


0.246E-2 

0.180E-2 


III  » 


UUULLLLLUU 


0.105E0 


0.277E-3 


2.762 

3.916 


18.821 


18.715 

19.878 


21.868 


25.787 


11 

UUULLLLLLUU 

0.105E0 

0.776E-4 

31.313 

12 

UlTULLLLLLLUU 

0.166E0 

0.526E-4 

34.991 

13 

UUULLLLLLULUU 

0.973E-1 

0.282E-4 

35.379 

14 

UUUULLLLLLULUU 

0.581E-1 

0.122E-4 

36.778 

TABLE 4 

Optimal  design  ofl-D  beam  structure,  to  minimise  "maximum”  strain  energy. 


Bay  No.:  Bay  in  which  the  optimal  minimum  “maximum”  non-dimensional  strain  energy  occurs 

Original  Energy:  Initial  “maximum”  non-dimensional  bay  strain  energy  of  the  periodic  structure. 

Final  Energy:  Non-dimensional  strain  energy  in  bay  N’  of  the  optimised  structure. 


No  of 

Optimal 

Original 

Final 

Bay 

Reduction 

Bays,  N 

Pattern 

Energy 

Energy 

No.,  N’ 

(dB) 

Q.=50 

9 

ULLLUUUUL 

0.667E-1 

0.297E-1 

1 

3.514 

0.296E-1 

1 

3.528 

10 

ULLLUUUULU 

0.540E  0 

12 

ULLLULUULULU 

0.404E  0 

0.297E-1 

1 

11.336 

0.296E-1 

1 

11.351 

40<Q<60 

9 

UUUUULULU 

0.486E  0 

0.710E-1 

1 

8.354 

0.449E-i 

1-2 

10.344 

10 

ULULULULLL 

0.606E  0 

0.643E-1 

1 

9.743 

0.45  IE- 1 

1-2 

11.283 

11 

ULULULULLUU 

0.456E  0 

0.682E-1 

I 

8.252 

0.425E-1 

1-2 

10.306 

12 

UUUIJUUUXJUUUL 

0.332E  0 

0.550E-1 

2 

7.808 

0.412E-1 

1-2 

9.062 

23<n<61 

9 

LLLLLLLLL 

0.234E  0 

0.203E  0 

1 

0.617 

0.979E-1 

1 

3.784 

n 

LLLLLLLLLL 

0.200E  0 

0.178E0 

1 

0.506 

■I 

0.95  IE- 1 

1-2 

3.228 

11 

UUUUUUUUULL 

0.198E0 

0.193E0 

1 

O.lll 

0.910E-1 

1-2 

3.376 

12 

UUUUUOUIJULUU 

0.314E0 

0.182E0 

I 

2.369 

0.803E-1 

1-2 

5.922 

1123 


Phase  £  AUenualion 


Figure  1;  A  simply  supported  periodic 


Itlijsj 


kinelic.enorgy  in  bay  12,  T, 


1128 


EFFECTS  OF  GEOMETRIC  ASYMMETRY  ON  VIBRATIONAL 
POWER  TRANSMISSION  IN  FRAMEWORKS 

J  L  Homer 

Department  of  Aeronautical  and  Automotive  Engineering 
and  Transport  Studies,  Loughborough  University 
Loughborough,  Leics  ,  LEll  3TU,  UK 


ABSTRACT 

Many  sources,  such  as  machines,  are  installed  on  supports,  or  frameworks, 
constmcted  from  beam-like  members.  It  is  desirable  to  be  able  to  predict 
which  wave  types  will  be  present  at  particular  points  in  the  support  structure. 
By  using  the  concept  of  vibrational  power  it  is  possible  to  compare  the 
contributions  from  each  wave  type.  Wave  motion  techniques  are  used  to 
determine  the  expressions  for  vibrational  power  for  each  of  the  various  wave 
types  present.  The  results  from  the  analysis  show  the  amount  of  vibrational 
power  carried  by  each  wave  type  and  the  direction  of  propagation. 
Consideration  is  given  to  the  effect  on  the  vibrational  power  transmission  of 
introducing  misalignment  of  junctions  in  previously  symmetric  framework 
structures.  By  splitting  a  four  beam  junction  in  to,  say,  a  pair  of  three  beam 
junctions  separated  by  a  small  distance,  it  is  possible  to  establish  the  effects  of 
separating  the  junctions  on  the  various  transmission  paths.  Unlike  other 
techniques  using  vibrational  power  to  analysis  frameworks,  the  model  keeps 
the  contributions  from  each  of  the  various  wave  types  separate.  This  allows 
decisions  to  be  made  on  the  correct  vibrational  control  techniques  to  be 
applied  to  the  structure. 


INTRODUCTION 

When  attempting  to  control  vibration  levels  transmitted  from  a  machine 
through  the  various  connections  to  the  structure  upon  which  it  is  mounted,  it  is 
desirable  to  be  able  to  identify  and  quantify  the  vibration  paths  in  the  stmcture. 
Often  large  machinery  installations  are  installed  on  frameworks  consisting  of 
beam  like  members.  These  frameworks  are  then  isolated  from  the  main 
structure.  Simple  framework  models  are  also  used  in  the  initial  design  stages 
of  automotive  body  shell  structures  to  determine  dynamic  responses. 

If  the  dominant  transmission  path  in  the  framework  is  identified  it  is 
possible  to  reduce  vibration  levels  by  absorbing  the  mechanical  energy  along 
the  propagation  path  in  some  convenient  manner.  By  utilising  the  concept  of 
vibrational  power  it  is  possible  to  quantitatively  compare  the  wave  type 
contributions  to  each  transmission  patL  In  order  to  predict  vibrational  power 
transmission  in  a  framework,  it  is  necessary  to  identify  the  wave  amplitude 
reflection  and  transmission  coefficients  for  each  joint  in  the  structure.  Lee  and 
Kolsky  [1]  investigated  the  effects  of  longitudinal  wave  impingement  on  a 
junction  of  arbitrary  angle  between  two  rods.  Similarly  Doyle  and  Kamle  [2] 
examined  the  wave  amplitudes  resulting  from  a  flexural  wave  impinging  on 
the  junction  between  two  beams.  By  using  the  reflection  and  transmission 
coefficients  for  different  joints,  it  is  possible  to  predict  the  vibrational  power 
associated  with  flexural  and  longitudinal  waves  in  each  section  of  the 


1129 


framework.  Previous  investigations  [3,4]  have  considered  the  effects  of  bends 
and  junctions  in  infinite  beams.  This  work  was  extended  to  consider  the  finite 
members  which  constitute  frameworks  [5].  Unlike  other  techniques  [6,  7] 
utilising  energy  techniques  to  analyse  frame-works,  the  technique  produces 
power  distributions  for  each  wave  type  present  in  the  structure.  By  comparing 
the  results  for  each  wave  type,  it  is  possible  to  apply  the  correct  methods  of 
vibration  control. 

The  technique  is  used  to  investigate  the  effect  of  geometric  asymmetry 
on  the  vibration  transmission,  due  to  steady  state  sinosodial  excitation,  in  a 
framework  structure  similar  to,  say,  those  used  in  the  automotive  industry 
(figure  1).  By  splitting  a  four  beam  junction  into  a  pair  of  three  beam 
junctions  separated  by  a  known  distance,  it  is  possible  to  establish  the  effect  of 
junction  separation  on  the  dominant  transmission  paths.  The  investigation 
presented  is  limited  to  one  dimensional  bending  waves  and  compressive  waves 
only  propagating  in  the  structure.  To  consider  the  addition  of  other  wave 
types  ie.  torsional  waves  and  bending  waves  in  the  other  plane,  the  analysis 
presented  here  for  the  junctions  should  be  extended  as  indicated  by  Gibbs  and 
Tattersall  [3]. 


TRANSMITTED  POWER  IN  A  UNIFORM  BEAM 

For  flexural  wave  motion,  consider  a  section  of  a  uniform  beam  carrying  a 
propagating  flexural  wave.  Two  loads  act  on  this  beam  element,  the  shear 
force  and  the  bending  moment.  It  is  assumed  that  the  flexural  wave  can  be 
described  by  using  Euler-Bemoulli  beam  theory,  so  that  the  displacement  can 
be  expressed  as 

W(x,t)  =  Af  sin  (cot-kfx), 

the  shear  force  acting  on  a  section  as 

S  =  Eia3W/ax3, 

and  the  bending  moment  on  the  section  as 
B  =  El  02  W/ax2. 

Then  the  instantaneous  rate  of  working  X  at  the  cross-section  is  given 
by  the  sum  of  two  terms  (negative  sign  merely  due  to  sign  convention). 

0t  0x0t  0X^  0X^  3x0t 

The  time  averaged  power 
{P)f  =  (1/T)  I  X  dt  then  is  given  by  (P)f  =  Elkf  coAf 

I  (1) 

For  longitudinal  wave  motion  consider  a  section  of  a  uniform  beam 
with  a  longitudinal  wave  propagating  through  the  beam 

U  (x,  t)  =  Ai  sin  (0)t-kix) 


1130 


The  instantaneous  rate  of  working  X  is  then 
X=  -EA(au/ax)u 


and  the  time  averaged  power  is 


Xdt  =  rEA0)kiA| 


(2) 


If  dissipation  is  present  in  the  structure,  the  modulus  of  elasticity  may 
be  considered  to  be  a  complex  quantity 

=  E(l  +  iTi) 


where  represents  the  loss  factor  of  the  material,  present  due  to 
inherent  material  damping. 

The  displacement  of  a  beam  at  a  distance  x  from  the  source,  due  to 
flexural  wave  motion  may  now  be  considered  to  be,  assuming  that 
material  damping  is  small. 


1  ^ 
-kri- 


W  =  Afe  4 

and  the  resulting  time  averaged  power  is  given  by 
(P}f  =  EIcok^e-'‘'’i2  Af 


(3) 


The  above  reduces  to  equation  (1)  at  the  source. 

Similarly,  the  displacement  of  beam,  due  to  longitudinal  wave  motion 
may  be  considered  to  be 

T  ^ 

-kiTi- 

U  =  Aie  ^  sin(cot-kix) 

and  the  resulting  time  averaged  longitudinal  power  may  be  rewritten  as 


(P),=iEAcok,e-‘''^’‘A? 


(4) 


WAVE  TRANSMISSION  THROUGH  A  MULTI  BRANCH  JUNCTION 

Consider  a  four  branch  junction  as  shown  in  figure  2.  Assuming  only  flexural 
and  longitudinal  waves  propagating  in  the  structure,  the  displacements  of  Arm 
1  will  be,  where  A4  represents  the  impinging  flexural  wave  arriving  from 
infinity. 


Wi(x,t)  =  (Aie^f’""  +  A3e‘^fi''  +  A4e  e‘“^ 


(5) 


1131 


(6) 


U,(x,t)  =  (Aae''‘‘i’‘)e‘“‘ 


Similarly  for  arms  2  to  4  the  displacement  will  be, 

where  ^  cos  0n  and  n  is  the  beam  number 

W„(v„,t)  =  )e‘“‘ 

(V) 

(8) 

Here  A3,  A4,  64^  are  travelling  flexural  wave  amplitudes',  Ai  and 
B2n  are  near  field  wave  amplitudes  and  Aa  and  are  travelling 
longitudinal  wave  amplitudes. 

In  previous  work  [2]  in  this  field  a  theoretical  model  was  used  in  which 
it  was  assumed  that  the  junction  between  the  beams  was  a  rigid  mass.  The 
mass  or  joint  is  modelled  here  as  a  section  of  a  cylinder.  This  represents  the 
physical  shape  of  most  joints  in  practical  systems.  It  has  been  shown  [4]  that 
the  joint  mass  has  an  insignificant  effect  on  the  reflected  and  transmitted 
power  for  the  range  of  values  used  in  this  work. 

The  joint  mass  Mj  =  pjTtL^J^/  4,  and  the  moment  of  inertia  of  the  joint 
isIj=ML2/8. 

By  considering  the  conditions  for  continuity  and  equilibrium  at  the 
beam  junction  the  following  expressions  may  be  written. 

For  each  arm 

For  continuity  of  longitudinal  displacement 

L  3W 

Ui=Un  COsen-WnSinen  +  -^^sinen 

For  continuity  of  flexural  displacement 

W,=u„sinen+W„cos0„-|  |^(l  +  cose„) 

For  continuity  of  slope 

8W;  ^  awn 


1132 


For  the  junction 
Equilibrium  of  forces 


'  '  2  '  '  3x2  J 


-vfp  fax  L32w„ 

1 1  I  aVS  2  3v|;^ 


El  Ai  ^  +  Mj  ^  =  i[E„  A„^cose„  +E„  I„^^sine„ 
3x  J  dt^  il^  3V„  3< 


E  T  a^W,  3  r„,  L3W| 

El  Ii  — t^  +  M;  — T  Wi-  — — — 


- ^  -I-  iVi ;  — y  1  “  “  ^T" 

3x2  2  3x 


n  ;^TI  ^  W 

=  Z  En  Ajj-  ”Sin6n-EnIn  2*^  COS0n 

1  I  5¥n 


WAVE  MOTION  AT  A  FORCED  OR  FREE  END 

As  indicated  in  figure  1 ,  the  framework  has  one  forced  end  and  one  free  end. 
Assuming  the  structure  is  only  excited  by  a  transverse  harmonic  force,  the 
boundary  conditions  are  as  follows: 

at  the  forced  end 


E«  =  Pe^«t 

3x2 


EI^  =  0 

3x2 


EA^  =  0 


Similarly  at  the  free  end  the  above  boundary  conditions  apply  with  the 
exception  that 

EI^  =  0 
3x2 


1133 


POWER  TRANSMISSION  THROUGH  A  FRAMEWORK 


The  structure  shown  in  figure  1  consits  of  one  four-beam  junction,  two  three- 
beam  junctions  and  four  two-beam  junctions.  From  the  equations  detailed  in 
the  above  two  sections,  it  is  possible  to  construct  matrices  of  continuity  and 
equilibrium  equations  for  sub  structures.  These  may  be  combined  to  obtain 
the  overall  matrix  for  the  system.  For  the  framework  shown  in  figure  1,  the 
size  of  the  overall  matrix  is  60x60.  This  matrix  may  be  solved  to  obtain  the 
sixty  unknown  wave  amplitude  coefficients  from  which  time  averaged 
transmitted  power  for  each  beam  may  be  calculated  using  equations  (3)  and 
(4). 

Normalised  nett  vibrational  power  is  then  calculated  at  the  centre  of 
each  beam  constituting  the  structure.  Nett  vibrational  power  may  be 
considered  to  be  the  difference  between  power  flowing  in  the  positive 
direction  and  power  flowing  in  the  negative  direction  for  each  wave  type. 
Normalised  nett  power  is  considered  to  be  nett  power  divided  by  total  input 
power.  The  input  power  to  a  structure  may  be  calculated  from  the  following 
expression  [8] 

Input  Power  =  ^  IFI IVI  cos0 


where  0  is  the  phase  angle  between  the  applied  force  and  the  velocity 
of  the  structure  at  the  forcing  position. 


Figures  3-6  show  the  nett  normalised  power  in  each  arm  of  a 
framework  structure  over  the  frequency  range  0-lkHz  excited  by  IN  force, 
whose  material  and  geometric  properties  are  given  in  Appendix  2.  For  the 


results  shown,  angle  1  is  45®  and  angle  x  is  40®  (or  the  ratio 


=  0.89)  and 


L  =  y  =  0.1m.  Using  these  parameters  the  ratio  of  the  length  of  beam  No.6  to 
beam  No.4  is  0.12.  The  predicted  flexural  power  is  shown  in  figures  3  and  4 
and  from  these  it  can  be  seen  that  the  dominant  transmission  paths  are  arms  1 
and  5,  the  forced  and  free  arms.  The  transmitted  power  in  arm  10  is  next 
dominant  and  comparable  to  arm  5  in  the  region  0-600Hz. 

The  response  for  ail  other  arms  are  small,  typically  less  than  5%  of 
input  power,  with,  as  would  be  expected,  arms  2  and  9  being  approximately 
identical  in  transmission  properties. 

Figures  5  and  6  show  the  nett  normalised  power  for  the  longitudinal 
waves  in  the  structure.  As  the  frequency  range  of  interest  corresponds  to  a 
flexural  Helmoltz  number  of  1  to  5  with  L  being  the  reference  length,  the 
conversion  of  power  from  flexural  to  longitudinal  waves  is  minimal.  From  the 
figures  it  can  be  seen  that  beams  1,  5,  6  and  10  have  identical  transmission 
characteristics,  which  would  be  expected  at  such  large  longitudinal 
wavelengths.  Significant  longitudinal  power  is  only  observed  in  arms  3  and  8 
in  the  frequency  region  200-300Hz.  This  frequency  region  coincides  with  a 
drop  in  the  flexural  power  due  to  the  structure  being  at  resonance  in  that 
region.  It  should  be  noted  that  power  transmitted  through  arms  3  and  8  has 
travelled  through  two  junctions. 


1134 


EFFECT  OF  GEOMETRIC  ASYMMETRY 


By  altering  the  ratio  of  angle  6i  to  angle  0x  it  is  possible  to  alter  the  length  of 
beam  6  and  hence  move  a  pair  of  three  arm  junctions  further  or  closer  apart. 
From  the  discussion  in  the  previous  section,  it  was  seen,  for  the  structure 
under  investigation,  that  the  dominant  flexural  path,  not  surprisingly,  is 
through  the  centre  of  the  structure,  whilst  the  peaks  in  longitudinal  power 

occur  in  beams  3  and  8.  Thus  0x  was  varied  and  the  effect  on  transmission  in 
the  dominant  paths  noted. 

Figures  7-9  show  flexural  power  for  arms  5  and  10  and  longitudinal 
power  for  arm  8  for  four  values  of  0x.  The  values  chosen  were  36°,  38.25°, 
40°  and  42.75°  which  are  equivalent  respectively  to  0x  over  0]  ratios  of  80%, 

85%,  90%  and  95%.  Thus  as  0x  increases,  the  structure  moves  to  being 
symmetrical  in  nature.  From  figure  7,  the  increase  in  junction  separation 
decreases  power  in  the  frequency  region  0-500Hz  and  increases  it  in  the  region 
500-lkHz.  In  beam  10  (figure  10)  the  effect  on  the  flexural  power  is  reversed 
with  increase  in  junction  separation  leading  to  increased  power  below  500Hz 
and  decreased  power  above  500Hz.  It  should  also  be  noted  that  increased 
junction  separation  has  little  effect  on  the  power  below  250Hz.  The  effect  was 
also  noted  on  all  other  beams  which  had  both  ends  connected  to  a  joint.  It 
may  be  concluded  that  at  long  flexural  wavelengths  the  junction  separation  has 
little  effect  with  the  impedance  mis-match  at  the  junctions  being  the  important 
criteria  to  effect  transmission.  It  should  also  be  noted  that  increasing  power  in 
one  arm  ie.  5,  causes  a  decrease  in  arms  ie.  10,  connected  to  it.  An  example  of 
the  effect  of  junction  separation  on  longitudinal  power  is  shown  in  figure  9. 
This  shows  nett  normalised  longitudinal  power  for  arm  8  for  the  same 

variation  in  0x.  Again  minimal  effect  is  seen  at  low  frequencies,  with 
increased  junction  separation  having  different  effects  in  different  frequency 
region.  Increasing  junction  separation  has  little  effect  on  the  region  between 
200  and  300Hz  when  the  longitudinal  power  was  dominant.  This  would  be 
caused  by  the  junction  separation  having  little  effect  on  the  structures  flexural 
natural  frequencies.  Only  by  shifting  those  would  the  peaks  in  longitudinal 
power  by  shifted  in  frequency. 


CONCLUSIONS 

Results  are  presented  for  normalised  nett  time  average  vibrational  power  for  a 
framework  structure.  The  geometric  symmetry  of  the  structure  is  broken  by 
allowing  one  angle  to  decrease  in  value.  The  effects  of  varying  the  angle 
change  by  up  to  20%  of  its  original  value  are  investigated.  Although  the 
results  presented  are  for  one  example  only,  highlighted  are  the  fact  that 
decreases  in  power  in  one  part  of  the  structure  result  in  increases  in  power  in 
another  part.  Also  shown  was  the  effect  of  splitting  a  junction  in  to  a  pair  of 
junctions  is  minimal  at  low  frequencies,  or  long  wavelengths.  From  the  results 
of  the  analysis  it  is  possible  to  establish  frequencies  and  positions  for 
minimum  power  on  the  structure.  Other  configurations  of  framework 
structure  may  be  analysed  by  applying  the  equations  presented. 


1135 


REFERENCES 


1.  J.  P.  LEE.  and  H.  KOLSKY  1972  Journal  of  Applied  Mechanics  39,  809- 
813.  The  generation  of  stress  pulses  at  the  junction  of  two  non-collinear 
rods. 

2.  J.  F.  DOYLE  and  S.  KAMLE  1987  Journal  of  Applied  Mechanics  54, 
136-140.  An  experimental  study  of  the  reflection  and  transmission  of 
flexural  waves  at  an  arbitrary  T-Joint. 

3.  B.  M.  GIBBS  and  J.  D.  TATTERS  ALL  1987  Journal  of  Vibration, 
Acoustics,  Stress  and  Reliability  in  Design,  109,  348-355.  Vibrational 
energy  transmission  and  mode  conversion  at  a  corner  junction  of  square 
section  rods. 

4.  J.  L.  HORNER  and  R.  G.  WHITE  1991  Journal  of  Sound  and  Vibration 
147,  87-103.  Prediction  of  vibrational  power  transmission  through  bends 
and  joints  in  beam-like  structures. 

5.  J.  L.  HORNER  1994  Proceedings  of  the  5th  International  Conference  on 
Recent  Advances  in  Structural  Dynamics,  SOUTHAMPTON  UK,  450- 
459.  Analysis  of  vibrational  power  transmission  in  framework  structures. 

6.  P.  E.  CHO  and  R.  J.  BERNHARD  1993  Proceedings  of  the  4th 
International  Congress  on  Intensity  Techniques,  SENLIS,  France,  347- 
354.  A  simple  method  for  predicting  energy  flow  distributions  in  frame 
structures. 

7.  M.  BESHARA  and  A.  J.  KEANE  1996  Proceedings  of  Inter-Noise  '96, 
LIVERPOOL,  UK  2957-2962.  Energy  flows  in  beam  networks  with 
complient  joints. 

8.  R.J.PINNINGTON  and  R.G.WHITE  1981  Journal  of  Sound  and  Vibration 
75,  179-197.  Power  flow  through  machine  isolators  to  resonant  and  non- 
resonant  beams. 


APPENDIX  1  -  NOTATION 


A 

-  Cross  sectional  area 

Q 

-  Axial  force 

Af 

-  Amplitude  of  flexural  wave 

s 

-  Shear  force 

Ai 

-  Amplitude  of  longitudinal  wave 

T 

-  Time  period 

B 

-  Bending  moment 

t 

-  Time 

E 

-  Young’s  modulus 

U 

-  Displacement  due  to 

E* 

-  Complex  Young’s  modulus 

longitudinal  wave  motion 

F 

-  Excitation  force 

V 

-  Velocity 

I 

-  Moment  of  inertia 

w 

-  Displacement  due  to 
flexural  wave  motion 

Ij 

Jw 

-  Moment  of  inertia  of  joint 

X 

-  Instantaneous  rate  of 
working 

-  Joint  width 

X 

-  Distance 

-  Loss  factor 

kf 

-  Flexural  wave  number 

0n 

-  Angle  of  Arm  n 

ki 

-  Longitudinal  wave  number 

Pj 

-  Joint  density 

L 

-  Joint  length 

M 

-  Moment  force 

Mj 

n 

P 

-  Joint  mass 

<}> 

Phase  angle 

-  Beam  number 

-  Transverse  force 

¥n 

-  Distance  along  Arm  n 

1136 


<P>f  -  Time  averaged  flexural  power  co  -  Frequency  (rad/s) 
<P>1  -  Time  averaged  longitudinal  power 

APPENDIX  2  -  MODEL  PROPERTIES 

33mm 
6mm 
5GN/m2 
1180kg/m3 
0.001 


1 


Beam  Breadth 
Beam  Depth 
Youngs  Modulus 
Density 
Loss  factor 


Figure  1 :  Framework  Structure 


iNett  Normalised  Power 


Nett  Normalised  Power 


Figure  5:  Longitudinal  Power  -  Beams  1-5 


(Beam  1 - ,  Beam  2 . ,  Beam  3  .  .  Beam  4 - ,  Beam  5 


Figure  6:  Longitudinal  Power  -  Beams  6-10 

(Beam  6 - ,  Beam  7 . ,  Beam  8  .  .  Beam  9 - ,  Beam  10 


1139 


THE  INFLUENCE  OF  THE  DISSff  ATION  LAYER  ON  ENERGY 
FLOW  IN  PLATE  CONNECTIONS 


Marek  Iwaniec,  Ryszard  Panuszka 

Technical  University  of  Mining  and  Metallurgy, 
Structural  Acoustics  and  Intelligent  Materials  Group 
30-059  Cracow,  al.  Mickiewicza  30,  Poland 


1.  Introduction 


Dynamic  behaviour  of  mechanical  strucmres  may  be  modelled  on  the  basis  of 
and  with  the  help  of  mathematical  apparams  used  in  Statistical  Energy  Analysis  (SEA) 
[5]  The  method  is  especially  useful  to  calculate  the  statistical  approach  vibroacoustical 
energy  flow  in  middle  and  high  frequency  range.  With  the  help  of  a  few  parameters, 
such'as-  modal  density,  damping  loss  factor,  coupling  loss  factor  and  the  value  of 
input  power,  building  linear  equations  set  it  is  possible  to  describe  the  flow  of 
vibroacoustical  energy  in  a  complicated  stmcture.  There  is  also  a  possibility  of  quick 
estimation  of  the  influence  of  constmction  method  on  the  vibroacoustical  parameters 
of  the  whole  set.  In  the  following  work  an  exemplary  application  of  one  of  the  most 
frequently  used  software  for  calculating  the  flow  of  acoustic  energy  has  been 
presented-  AutoSEA  programme  [1].  The  aim  of  the  work  is  practical  modelling  of 
vibroacoustical  energy  flows  through  screw-connection  of  two  plates  and  comparing 
quantity  results  with  experimental  (outcome)  measurements.  Equivalent  coupling  loss 
factor  has  been  calculated  for  a  group  of  mumally  combined  elements  constimting  a 
construction  fragment.  A  comparison  between  the  measured  results  and  the  value  of 
coupling  loss  factor  in  linear  joint  (e.g.  in  welded  one)  has  also  been  made.  Using  the 
method  of  fmite  elements,  the  influence  of  rubber  separator  thickness  on  the  value  of 
the  first  several  frequencies  of  free  vibrations  has  been  computed  as  well. 

2.  Physical  model  of  plate  connection 

A  connection  of  two  perpendicular  plates  has  been  chosen  for  modelling  the 
flow  of  vibroacoustical  energy  in  mechanical  joints.  Connection  diagram  is  presented 
in  Figure  1.  On  the  length  of  common  edge  the  plates  has  been  joined  with  anglesteel 
by  screws.  A  rubber  separator  (4)  has  been  placed  between  the  excited  plate  and  the 
anglesteel  leg  (3). 


1143 


Modelled  stmcture 


3.  SEA  model 

In  order  to  carry  out  the  vibroacoustical  analysis  of  the  system  using  Statistical 
Energy  Analysis  a  model  of  the  examined  strucmre  has  been  built.  It  has  been 
assumed  that  in  every  element  of  the  construction  only  flexural  waves  propagate. 
Every  plate  and  the  rubber  layer  have  been  modelled  with  just  one  appropriately 
chosen  subsystem.  The  anglesteel,  however,  has  been  modelled  as  a  continuous 
connection  of  two  plates  having  the  dimensions  which  correspond  to  the  anglesteel 
legs  the  plates  themself  being  set  at  the  right  angle. 


Fig.  2.  SEA  model  of  a  system  Fig-  3.  Modal  densities  of  the  subsystems 


1144 


Using  the  SEA  method  we  are  able  to  describe  the  flow  of  the  vibroacousdcal 
energy  in  middle  and  high  frequencies  with  an  algebraic  equation  set.  The  exMined 
system  consists  of  five  simple  subsystems,  of  which  only  one  is  exited  to  vibration 
with  applied  force.  The  flow  of  vibroacoustical  energy  m  the  model  presented  is 
depicted  with  the  following  equation: 


^  1  tot 

-^21 

0 

0 

12 

^  2  tot 

32 

0 

0 

“'•123 

^  3  tot 

-‘n43 

0 

0 

“''134 

h  4  tot 

0 

0 

0 

"'’145 

0 

tot 

w, 

0 

^2  tot 

0) 

0 

0 

• 

^3  tot 

= 

0 

-^54 

^4  tot 

0 

^  5  tot 

^5  tot 

0 

^  ^  4-  is  a  total  coefficient  of  energy  loss  for  every  subsystem, 

rj’.  '"'  -  intemarioss  factor  of  the  subsystem, 

ri..  -  coupling  loss  factor  between  subsystems, 

E-  -  the  mean  vibrational  energy  in  Af  frequency  band  in  i-subsystem, 
W  -  the  input  power  carried  into  i-subsystem  from  outside. 


To  determine  the  elements  of  the  coefficients  matrix  in  eq.  1  it  is  necessary  to 
know  [1,4,6]  coupling  loss  factors  (CLF)  between  structural  subsystems  and  damping 

In  the  SEA  model  in  question  transmission  of  the  acoustic  energy  occurs  in  two 
tvpes  of  connections  between; 

the  plate  and  the  beam  (the  point  joint  of  the  beam  which  is  parallel  to  the  edge 

of  the  plate,  (transfers  flexural  waves),  .  .  „ 

two  plates  (linear  connections  and  point  joints  transferring  flexural  waves). 


The  coupling  loss  factor  between  the  plate  and  the  beam  which  vibrate  in  the 
flexural  way  (in  the  case  of  the  point  joint),  is  defined  with  following  equation  [1]. 


1.75c^ 


(2) 


where: 

c  -  is  the  speed  of  flexural  wave, 

T  -  transmission  factor, 

Q  -  the  number  of  point  connections, 
Gj  -  angular  frequency, 

A  -  the  surface  of  the  plate. 


1145 


The  speed  of  the  flexural  wave  in  the  first  plate  c„  can  be  calculated  in  the 
following  way: 


(3) 


In  the  model  under  examination  formula  (2)  defines  coupling  loss  factors  between  the 
anglesteel  leg  (2)  and  the  beam  (4)  -  ( factors  1,3.  and  tiJ  or  the  beam  (4)  and  the  plate 
(5)  -  rj45  and  7/54. 

The  flow  of  energy  between  two  plates  (which  are  connected  at  the  right  angle 
and  which  vibrate  in  flexural  way)  is  defined  with  the  following  formula  in  the  case 
of  linear  connection: 


where: 

1  -  is  the  length  of  the  connection. 

With  above  formula  it  is  possible  to  describe  the  flow  of  energy  through  correctly 
made  welded  joints  of  plates  or,  for  example,  through  bent  plates.  In  the  system 
presented  the  factor  determines  the  flow  of  energy  between  anglesteel  legs  (2)  and 

(3) 

The  coupling  loss  factor  between  two  plates  with  a  point  joint  is  described  by 
the  following  formula: 


iiL^ 

3  (o.X, 


(5) 


This  type  of  connection  occurs  between  the  plate  representing  the  anglesteel 

leg  and  the  plate  (5).  ^  . 

After  defining  the  value  of  factors  matrix  in  the  first  equation  it  is  possible  to 

specify  the  ratio  of  the  vibroacoustical  energy  gathered  in  plate  (1)  and  (5). 

_  (n  2tof^  3t0t~  ^  24^  32)  4fot^  5tnt~  ^  54^  45^  ~  34^  43^  5tot  (5) 

£5  ^21^  32'^  43^  54 

The  damping  loss  factor  is  important  parameter  of  every  subsystem.  For  steel 
plates  used  in  the  experiment  the  value  of  the  damping  loss  factor  have  been  measured 
experimentally  with  the  decay  method.  The  results  of  the  measurements  have  been 
presented  in  figure  4.  The  frequency  characteristic  of  rubber  damping  has  been  shown 
in  figure  5. 


1146 


DLF  of  steel  [-] 


frequency  [Hz]  frequency  [Hz] 


Fig.  4.  Damping  loss  factor  of  steel  Fig.  5.  Damping  loss  factor  of  rubber 
4.  Experimental  research 

Experimental  investigations  have  been  carried  out  for  connections  made  with 
the  use  of  rubber  separator  (elastic  layer)  of  50°  Shore  hardness.  The  connection  was 
build  up  of  two  identical,  perpendicular  plates  connected  each  to  another  using  the 
anglesteel  and  the  elastic  rubber  layer.  These  are  the  properties  and  material 
parameters  of  individual  elements: 
plates: 

-  material  constructional  steel  (St3); 

-  dimensions:  500  *  500  *  2.2  mm; 

-  Young  modulus:  2,1  10'^  Pa. 
anglesteel: 

-  material:  constructional  steel  (St3); 

-  dimensions:  L  40  40  2.2  mm; 

-  Young  modulus:  2,1  10' ‘  Pa 
elastic  layer: 

-  material:  rubber  50°  Shore 

-  dimensions:  500  40  mm 

-  thickness:  2,  3,  4,  5,  6  mm 

To  avoid  the  loss  of  mechanical  energy  in  the  environment,  during  measure¬ 
ments  the  construction  was  suspended  to  the  supporting  frame  with  three  weightless 
strings  in  such  way  that  only  rigid  body  motions  in  the  plane  perpendicular  to  the  plate 
surface  can  occur.  The  excitation  of  the  wide-band  type  with  constant  power  spectral 
density  was  applied  in  the  symmetry  axis  of  the  plate  (5)  about  20  mm  below  the 
upper  edge  (Figure  1).  During  the  experiment  the  distribution  of  vibrating  velocities 
on  the  plate  surface  was  obtain  by  non  contact  method  using  laser-vibrometry. 


1147 


5.  FEM  model 

Vibration  of  modelled  structure  in  low  frequency  has  been  analyzed  by  toe 
Finite  Element  Method  (FEM).  Several  FEM  models,  was  build  in  order  to  consider 
L  valid  thickness  of  the  mbber  layer.  The  mbber  thickness  has  been  from 

0  mm  to  6  mm.  The  dimensions  and  material  parameters  of  the  plates  and  the 
LSesteel  was  constant.  The  structure  was  fixed  in  four  comers.  Calc^auons  were 
mfde  for  a  division  of  the  strucmre  into  608  elements  of  type  BRI^8.  The  mfluence 
of  the  rubber  thickness  on  the  eigenfrequencies  was  remarkable.  The  results  for  first 
15  eigenfrequencies  are  presented  in  the  table  1 .  In  fig.  was  shown  also  the  changes 
of  the  value  of  natural  frequencies  in  comparison  with  the  natural  frequencies  of 
strucmre  without  applying  the  mbber  layer. 


table  1.  Namral  frequencies  of  connection 


LaDlc  i.  i>aLutai  ai.  —  - 

1  Natural  frequencies  (Hz)  for  various  rubb 

■  ■ .  -  -  • 

er  layer  tlrickness 

0  nun 

2  nun 

3  mni 

4  nun 

5  nun 

127.8 

123.1 

121.5 

119.4 

117.5 

297.5 

235.8 

224.6 

212.6 

204.6 

507.6 

432.5 

391.3 

357.1 

330.0 

1003.2 

566.9 

525.7 

503.7 

491.9 

1282.3 

807.1 

768.9 

733.5 

712.2 

1900.5 

1185.2 

1169.4 

1123.2 

1070.2 

2386.7 

1570.2 

1483.6 

1376.2 

1304.6 

3272.3 

1851.6 

1634.2 

1531.6 

1490.8 

3603.4 

2303.1 

2263.4 

2167.8 

2147.3 

4263.3 

2611.2 

2539.5 

2496.9 

2463.7 

4759.5 

3192.7 

3515 

3110.1 

2913.2 

5438.4 

3926.7 

3874.6 

3626.4 

2937.0 

7627.4 

4110.1 

4105.5 

3639.9 

3026.6 

7691.6 

5070.7 

5137.0 

3670.5 

3111.2 

11543.2 

5547.1 

5503.9 

3721.9 

3181.2 

The  decreasing  of  the  absolute  values  of  the  natural  frequences  is  observed  according 
to  increasing  of  the  thickness  of  the  rubber  layer.  Beginning  from  die  third  of 

°ib  adons  of  the  system  the  decrease  of  the  natural  frequencies  is  almost  constant  for  the 
"layer  2  mm  or  3  mm  and  is  continuously  decreasing  for  rubber  layer  4-6  mm  (See 

fig.  6). 


1148 


Fig.  6.  Changes  of  natural  frequencies 

6.  Comparison  with  experimental  results 

The  equivalent  coupling  loss  factor  defining  the  energy  flow  between  the  plates  has 
been  determined  experimentally  [31  for  a  model  consisting  of  two  subsystems;  two  plates. 
The  coupling  loss  factor  in  such  two  element  model  may  be  specified  by  the  following 

equation: 


1  zast. 


^2  ^1101 

N,  E2.0C 


-  1 


(7) 


The  quotient  of  plate  energies  E,/E,  in  a  two-element  model  is  relevant  to  the  quotient  of 
energy  in  the  first  and  fifth  subsystem  (E./E,)  in  the  five  element  model  presented  m  figure 
0  the  value  of  these  quotient  is  defined  with  the  equation  (6) 

In  the  picture  we  have  presented  the  values  of  the  equivalent  coupling  loss  factor  in 
the  connection.  Individual  points  in  the  diagram  show  the  results  obtained  experimentally. 
The  values  received  in  computer  simulation  have  been  presented  as  a  continuous  diagram. 
The  upper  curve  shows  the  values  of  the  coupling  loss  factor  in  the  joint  before  the 
application  of  the  rubber  layer. 

In  the  frequency  range  above  ca.  125  Hz  we  have  received  a  very  good  comparison 
of  experimental  results  and  computer  simulation  results  performed  with  the  AutoSEA 
software  At  the  frequency  of  about  200Hz  there  occurs  a  local  minimum  of  the  equivalent 
coupling  loss  factor  between  the  plates.  The  value  of  the  minimum  is  essentially  influenced 
by  the  value  of  the  rubber  damping  loss  factor.  The  frequency  (with  the  minimum  CLF)  is 
strongly  influenced  by  the  peak  frequency  of  the  rubber  damptng  curve. 


1149 


o.ooo2ii - 


symulation 


experiment 


welded  plates 


16'  ‘  '40'  '  ’lOO  250  630  1600  4000 

frequency  [Hz] 

Fig.7  Equivalent  CLF  factor  in  the  joint 

7.  Conclusion 

A  way  of  modelling  the  vibroacoustical  energy  flow  with  the  help  of  SEA  method  has 
been  presented  in  the  work.  We  have  examined  the  screw  connection  of  two  plates,  where 
a  rubber  elastic  layer  has  been  applied,  A  comparison  has  also  been  made  between  the  results 
of  computer  simulation  of  the  mechanical  energy  flow  with  SEA  method  and  the  experimental 
results,  and  thus  we  have  noticed  the  good  correlation,  especially  as  far  as  middle  frequencies 

The  joint  modification  through  introduction  of  the  rubber  separator  has  a  remarkable 

impact  on  the  acoustic  energy  flow:  ,  ^ 

-  The  application  of  the  elastic  layer  in  the  Joint  in  question  lowers  the  value  ot 

eauivalent  coupling  loss  factor  in  the  whole  frequency  range. 

-  the  value  of  rubber  damping  factor  has  most  significant  influence  on  the  acoustic 

eneroy  flow  through  connection  in  middle  frequencies  range: 

“  The  minimum  value  of  the  equivalent  coupling  loss  factor  in  a  joint  is  essentially 
influenced  by  the  rubber  separator  damping  loss  factor. 

*  The  frequency  of  minimum  CLF  occurrence  is  strongly  influenced  by  the  peak 
frequency  of  the  rubber  damping  curve. 

The  increasing  thickness  of  the  rubber  layer  produces  on  decreasing  natural  frequencies  of 
the  structure. 


8.  Bibliography: 

III  AutoSEA  -  User  Guide  Vibro-Acoustic  Sciences  Limited  1992. 

121  Fahy  F.J.  Sound  and  Structural  Vibration;  Radiation,  Transmission  and  Response  Academic  Pres 
nTkirtuin  J  Smals  N.  Panuszka  R.  "Method  of  estimating  the  coupling  loss  factor  for  a  set  of 

nlates"  Mechanic,  Technical  University  of  Mining  and  Metallurgy,  Cracow,  10,1991. 

[^41  Lalor,  N.:  The  evaluation  of  SEA  Coupling  Loss  Factors.  Proc.  V  School  Energy  Methods  in 

Vibroacoustics"  -  Supplement,  Krakow-Zakopane  1996.  ,  .  „  „ 

[51  Lyon  R.,  DeJong  R.;  Theory  and  Application  of  Statistical  Energy  Analysis.  Butterworth- 

Heinemann,  Boston,  1995. 


1150 


Variation  Analysis  on  Coupling  Loss  Factor 
due  to  the  Third  Coupled  Subsystem  in 
Statistical  Energy  Analysis 

Hongbing  Du  Fook  Fah  Yap 
School  of  Mechanical  &  Production  Engineering 
Nanyang  Technological  University 
Singapore  639798 


Abstract 

Statistical  Energy  Analysis  (SEA)  is  potentially  a  powerful  method  for 
analyzing  vibration  problems  of  complex  systems,  especially  at  high  frequen¬ 
cies.  An  impoitant  parameter  in  SEA  modeling  is  the  coupling  loss  factor 
which  is  usually  obtained  analytically  based  on  a  system  with  only  two  cou¬ 
pled  elements.  Whether  the  coupling  loss  factor  obtained  in  the  classical  way 
is  applicable  to  a  practical  problem,  which  normally  comprises  of  more  than 
two  elements,  is  of  importance  to  the  success  of  SEA.  In  this  paper,  the  varia¬ 
tion  of  coupling  loss  factor  between  two  subsystems  due  to  the  presence  of  a 
third  coupled  subsystem  is  investigated.  It  is  shown  that  the  degree  to  which 
the  coupling  loss  factor  is  affected  depends  on  how  strong  the  third  subsystem 
is  coupled.  It  also  depends  on  the  distribution  of  the  modes  in  the  coupled  sub¬ 
systems.  This  kind  of  effect  will  diminish  when  the  damping  is  high,  subsys¬ 
tems  are  reverberant,  or  ensemble-average  is  considered,  but  not  for  individual 
cases. 


1  Introduction 

SEA  is  potentially  a  powerful  method  for  analyzing  vibration  and  acoustic  problems 
of  complex  systems,  especially  at  high  frequencies,  because  of  the  simplicity  of 
its  equations  compared  to  other  deterministic  analysis  techniques.  SEA  models  a 
system  in  terms  of  interconnected  subsystems.  The  coupling  parameter  between 
any  two  subsystems  is  characterized  by  a  coupling  loss  factor.  If  the  coupling  loss 
factors  and  internal  (damping)  loss  factors  of  all  subsystems  are  known,  the  power 
balance  equation  (e.g.,  see  [1])  for  each  subsystem  can  be  established.  From  this  set 
of  equations,  SEA  predicts  the  system  response  (due  to  certain  types  of  excitation) 
in  terms  of  the  average  energy  of  every  subsystem.  The  energy  can  in  turn  be  related 
to  other  response  quantities  such  as  mean  velocity  or  strain. 


1151 


Historically,  the  SEA  power  balance  equations  were  initially  derived  from  an 
analysis  of  two  coupled  oscillators  [2,3].  It  has  been  shown  that  the  energy  flow 
between  them  is  directly  proportional  to  the  difference  in  their  uncoupled  modal 
energies.  The  theory  has  then  been  extended  to  systems  with  multi-coupled  sub¬ 
systems  (e.g.,  [4]).  Strictly  this  extension  is  only  applicable  if  certain  assumptions 
are  justified  [4, 5].  Also  the  new  concept  of  indirect  coupling  loss  factor,  which 
is  used  to  represent  the  energy  flow  proportionality  between  the  indirectly  coupled 
subsystems,  is  also  introduced. 

In  practice,  the  indirect  coupling  loss  factors  are  normally  ignored  in  SEA  ap¬ 
plications  because  they  are  very  difficult  to  determine  analytically.  Only  coupling 
loss  factors  between  directly  coupled  substructures  are  considered.  Some  analy¬ 
ses  [6-8]  have  shown  that  this  approximation  may  lead  to  significant  errors  in  the 
predicted  results  if  certain  conditions  are  not  met  in  the  system.  These  conditions 
include  not  only  the  well  known  requirement  of  weak  coupling  between  subsystems 
(e.g.,  see  [6]),  but  also  others,  such  as  given  by  Langley  that  the  response  in  each 
element  must  be  reverberant  [7];  and  by  Kean  that  there  should  be  no  dominant 
modes  (peaks)  inside  the  frequency-averaging  band  [8].  However,  it  is  usually  dif¬ 
ficult  to  know  whether  these  conditions  are  satisfied  for  a  particular  system.  In  fact 
the  above  mentioned  conditions  do  not  always  hold  for  practical  engineering  cases. 
On  this  point  of  view,  the  importance  of  a  coupling  loss  factor  for  describing  the 
coupling  between  indirectly  coupled  subsystems  are  to  be  further  examined. 

A  related  question  is  whether  the  coupling  loss  factors  obtained  from  the  system 
with  only  two  subsystems  can  still  be  applied  when  other  subsystems  are  present. 
Generally,  the  coupling  loss  factor  is  sensitive  to  the  amount  of  overlap  between 
the  modes  of  the  two  coupled  subsystems.  When  additional  subsystems  are  cou¬ 
pled  to  the  original  two-subsystem  model,  the  mode  distributions  of  the  originally 
coupled  two  subsystems  will  be  affected.  The  change  of  mode  distributions  will  fur¬ 
ther  affect  the  modal  overlap  between  the  coupled  two  subsystems  and  finally  the 
coupling  loss  factor  between  them.  However,  general  estimation  methods  for  cou¬ 
pling  loss  factor  assume  that  the  coupling  parameters  between  two  subsystems  are 
not  affected  much  by  the  presence  of  the  other  subsystems.  Therefore  the  conven¬ 
tional  approaches  of  deriving  coupling  loss  factor  are  mostly  based  on  consideration 
of  a  two-subsystem  model  only.  One  method  is  the  wave  approach,  by  which  the 
coupling  loss  factor  used  in  the  SEA  applications  are  derived  analytically  from  aver¬ 
aged  transmission  factors  of  waves  that  are  transmitted  through  a  Junction  between 
semi-infinite  subsystems.  This  method  only  takes  into  account  local  properties  at 
the  joints  and  sometimes  may  be  inaccurate.  Recent  research  [1,7,9-11]  based 
on  the  model  with  two-coupled  subsystems  has  shown  that  the  coupling  parameter 
does  depend  on  other  system  properties,  such  as  damping  loss  factor,  etc.  It  can  be 
argued  that,  if  there  is  a  third  coupled  subsystem,  the  coupling  parameters  between 
the  first  two  subsystems  will  also  depend  on  the  energy  flow  to  the  third  subsys¬ 
tem.  Therefore,  from  a  practical  point  of  view,  the  coupling  loss  factor  estimated 
for  two-coupled  subsystems,  ignoring  the  indirectly  coupled  subsystems,  can  only 
be  of  approximate  value. 


1152 


In  this  paper,  the  variation  of  coupling  loss  factor  between  two  subsystems  due 
to  the  presence  of  a  third  coupled  subsystem  is  studied.  In  the  following  sections, 
the  coupling  loss  factor  is  firstly  expressed  in  terms  of  global  mobility  functions. 
The  exact  solution  of  mobility  functions  is  only  for  simple  structures.  However,  for 
general  structures,  it  can  be  obtained  by  Finite  Element  Analysis  (FEA)  [10, 1 1]. 
The  coupling  loss  factors  obtained  respectively  in  the  cases  with  and  without  the 
third  subsystem  in  the  model  are  compared  for  two  particular  system  configura¬ 
tions,  respectively.  The  system  used  in  this  investigation  is  one-dimensional  simply 
supported  beanis  coupled  in  series  by  rotational  springs.  By  varying  the  spring  stiff¬ 
ness,  the  strength  of  the  coupling  between  the  second  and  the  third  subsystems  can 
be  changed.  It  is  shown  that  the  effect  of  the  third  coupled  subsystem  on  the  cou¬ 
pling  loss  factor  between  the  first  two  coupled  subsystems  depends  on  how  strong 
the  third  subsystem  is  coupled.  For  each  individual  case,  it  is  also  shown  that  this 
kind  of  effect  may  be  positive  or  negative,  depending  on  the  distribution  of  modes 
in  the  coupled  subsystems. 


2  Coupling  Loss  Factor  by  Global  Modal  Approach 


In  this  section,  a  modal  method  is  used  to  derive  coupling  loss  factor  in  a  sys¬ 
tem  with  any  number  of  coupled  subsystems.  The  result  is  then  simplified  for  two 
cases:  (1)  a  three-subsystem  model;  (2)  a  two-subsystem  model  which  is  simply 
substructured  from  the  previous  three-subsystem  model  by  disconnected  the  third 
subsystem. 

For  a  linear  system  which  consists  of  N  coupled  subsystems,  if  “rain-on-the- 
roof”  excitation  [10]  is  assumed  to  be  applied  to  each  subsystem  in  turn,  the  corre¬ 
sponding  response  energy  can  be  expressed  as 

I  I  drdsdu,  (1) 

sith.'ii/sl.Kmi  Mii.bsy.'it.emj 

where  /?/;  is  the  total  time-averaged  response  energy  of  the  subsystem  i.  due  to 
the  excitation  on  the  subsystem  j,  is  the  transfer  mobility  function  be¬ 

tween  the  response  points  r  and  the  excitation  point  H  is  the  averaging  range 
of  frequency,  u.',  in  and  S  are  the  mass  density  and  the  power  spectral  density  of 
excitation.  The  input  power  due  to  the  excitation  is  given  by 

I  RelH(s,.i,L0)]d.sdu  (2) 

Q  sahsyale'inj 


where  is  the  real  part  of  the  point  mobility  at  the  position 

simplicity,  two  terms,  a/,  and  are  defined  as 


a,,  = 


i>,  =  ^  = 


Al 

m;S  j 

Ik 


\  H [r,  s,uj)\~  dr  ds  cl.in 


n  sii.bsy.'iUjvii  /iubfiyslenij 

lle[H{.^,s,ij)]  dsd.u: 

n  subsyslr;mj 


For 


(3) 

(4) 


1153 


The  mobility  function,  H{r,.s,uj),  is  to  be  expressed  in  terms  of  the  global  modes 
of  the  system,  which  can  be  obtained  by  Finite  Element  Analysis  (FEA).  By  the 
principle  of  reciprocity  of  the  mobility  function,  the  relation  of  aij  =  a  ji  always 
holds  regardless  of  the  strength  of  coupling  and  the  magnitude  of  input  power  if  the 
excitation  is  “rain-on-the-roof Theoretically  applying  the  Power  Injection  Method 
[12]  we  can  obtain  the  SEA  equation  as 

n  =  [77]E  (5) 

where  H  =  {Hi,  ila,  -  •  •  ,  and  E  =  {E[,  E-z,  -  -  -  ,  E^r}'^ .  The  SEA  loss 
factor  matrix  [77]  is 


iVl  + 


i.^2 


(7?yV  + 


■niiau 

niiCLiisr 

bi 

b-z 

bN 

■nua-ii 

■nizCL-zN 

bi 

bo 

bjV 

77?.;Va,Yi 

•m.A/a/VY 

^1 

Ih 

b^! 

=  —  B  A“'M- 

UJr 


where,  //;  is  the  internal  loss  factor  for  subsystem  i.,  v/,,  is  the  coupling  loss  factor 
from  subsystem  -I.  to  subsystem  j,  ujc  is  the  central  frequency  of  the  averaging  band 

n. 


A  =  [a,,] 


From  equation  (6),  the  reciprocity  principle  of  the  coupling  loss  factors  can  be  easily 
seen,  due  to  aij  =  aji. 


1 

''■bi 

,  M  = 

L  ’-J 

Vij  n?. ;  b  j  ni  jbj/ (cu^/i  /  2) 

i]ji  niibi  ■niibi/iujc'^l'l) 


Where,  (77?.;6,:)/(u,v7r/2)  is  the  averaged  real  part  of  point  mobility  [4]  and  can  be 
regarded  as  the  generalized  modal  density  of  the  subsystem  i.  Assuming  weak 
coupling  and  light  damping,  it  approximately  equals  to  the  classical  definition  of 
modal  density  [13].  Therefore,  the  relation  given  by  equation  (7)  also  reduces  to  the 
classical  reciprocity  principle. 


1154 


2.1  Substructured  two-subsystem  model 


Figure  1 :  A  general  SEA  model  with  three  coupled  subsystems  in  series 


Consider  a  whole  system  with  three  substructures  coupled  in  series  as  shown  in 
Figure  1 .  If  subsystem  3  is  removed,  the  coupling  loss  factors  between  subsystems 
1  and  2  are  given  in  the  equation 

biCL22  bi(Li-2 

"h '^712  — ^?2L  _  ^  n?. i(fi  1  [fi22  —  '^7.2(^11^22  —  <'^i2^^'2l) 

—  7/12  772+7/21  u-V  _ ^20.21 _  _ ^2(1 1 1 _ 

777 i{an<^f22  “  <^^12f''2l)  7772(^110-22  —  O12O21) 

r  bj  ^to-12  1  (8) 

^  J_  777  ifli|  7772011022 

62O21  ^2 

7771O11O22  777.2  0  22 

The  approximation  in  the  above  equation  is  due  to  011022  »  012O21  when  the  cou¬ 
pling  is  weak.  Manipulating  equation  (8)  with  or  without  using  the  approximation 
both  can  work  out  the  coupling  loss  factors  7/12  and  7/21  as 


(9) 


r  61  6-, 

( 777  1  I  )  ( 77  72  62  )  - 7/2 - 77 

07^-777-10 1  1  07^777.2022 

!  ,  ^2  .  f.  *^1 

77  7  20'7 -  777  [Oi 

"07c7772022  07^-777-1  011 


(10) 


The  equations  are  true  regardless  of  the  strength  of  the  coupling.  It  can  be  seen  that 
7/12  and  7/21  depend  on  the  values  of  the  three  terms  777.,+/,  7//  and  bij{u:^ni-,au).  The 
first  two  are  the  generalized  modal  density  and  the  internal  loss  factor,  or  in  combi¬ 
nation  equivalent  to  modal  overlap  factor.  The  third  one,  by  noting  the  definitions 
of  6;  and  an,  is  the  ratio  of  input  power  to  response  energy  for  the  directly  excited 
subsystem,  i.e.,  the  total  loss  factor  of  subsystem  i.  From  equation  (8),  this  term 
can  be  approximately  expressed  as 


Total  loss  factor 
of  subsystem  i 


b; 

miLOcCLii 


m  +  5I'+-.7 


(11) 


1155 


In  the  classical  wave  approach,  where  semi-infinite  subsystems  are  assumed,  the 
total  internal  loss  factors  becomes 

■n  total,  I  =  m  +  Vn  and  l]iot.al,2  =  112  +  vTi  (^2) 

where  is  the  classical  coupling  loss  factor.  Substituting  equation  (12)  into  equa¬ 
tions  (9)  and  (10),  i]ij  reduces  to  the  classical  iiff,  which  only  depends  on  the  local 
properties  at  the  joints  rather  than  other  properties  of  the  system,  such  as  damping. 
If  the  subsystem  modal  parameters  are  used  to  evaluate  the  term,  bil{ujcmiaii),  then 
the  total  loss  factor  is  Just  the  internal  loss  factor  of  the  subsystem  and  the  cou¬ 
pling  loss  factor  is  equal  to  zero.  This  is  reasonable  because  using  the  uncoupled 
modal  parameters  instead  of  the  coupled  modal  parameters  is  actually  equivalent  to 
removing  the  coupling  between  two  subsystems. 

However,  for  finite  system  where  the  assumption  of  semi-infinity  is  not  justified, 
there  will  be  no  immediate  simplication  for  equations  (9)  and  (10).  Numerically, 
FEA  can  be  employed  to  obtain  the  global  modes  and  then  the  coupling  loss  factor 
can  be  calculated  [10, 1 1]. 


2.2  Full  three-subsystem  model 

Instead  of  substructuring,  consider  the  three-subsystem  model  as  a  whole  system, 
shown  in  figure  1.  Now  the  order  of  equation  (6)  is  reduced  to  3.  With  the  global 
modal  parameters  obtained  from  FEA,  the  coupling  loss  factors  can  be  directly 
evaluated.  However,  when  the  coupling  between  subsystems  is  weak,  the  order- 
reduced  equation  (6)  is  still  able  to  be  simplified.  Matrix  A  may  be  alternatively 
expressed  in  the  form  of 


■  a,  L 

0 

0  ‘ 

■  0 

«L2 

0 

■  0 

0 

«13  ' 

A  - 

0 

(t-ll 

0 

-f 

a\2 

0 

d.23 

+ 

0 

0 

0 

0 

0 

«33  . 

0 

((•23 

0  . 

.  (l.[3 

0 

0 

where  the  terms  on  the  right  side  are  sequentially  defined  as  Ai,  A2  and  A3 .  Under 
the  assumption  of  weak  coupling,  the  non-zero  entries  in  Ai,  A2  and  A3  will  be  of 
the  order  O(t^),  (9(e^)  and  respectively  [14],  The  inverse  of  matrix  A  may 

be  approximately  written  as 

=  Ai”^  —  Ai“^ A2 Ai”^  —  Ai"^ A3 Ai  ^  4- Ai  ^A2Ai  ^A2Ai 


+  ••• 


(14) 


As  an  approximation,  substituting  only  the  first  three  terms  in  equation  (14)  into 
equation  (6)  gives 


[tj]  ^  l/uv- 


/>■ 

bo  f  t’ 12 
rn  I  f(  1 1  a  2  2 
b3(l-\3 

ni[a\  KM. 3 


■ni2a  1 1  (1-22 

b2 

1112(1.22 

63^23 

1112(1.22^(33 


bi(t.\3 
in3(l\\(l.33 
bid.  13 
in.3Ct.2'i((33 

63  ~ 

1((3((33 


(15) 


1156 


Generally,  under  the  condition  of  weak  coupling,  the  indirect  coupling  loss  factors, 
■ihs  and  7731,  are  much  smaller  than  the  direct  coupling  loss  factors  and  the  internal 
loss  factors  [5,7, 15].  The  diagonal  elements  in  [rj]  can  therefore  be  approximated  to 
the  sum  of  internal  loss  factor  and  direct  loss  factor.  It  can  be  shown  from  equation 
(15)  that  equation  (11)  remains  valid  for  three  coupled  subsystems.  But  in  the  three- 
subsystem  case,  the  term,  b,|{uJcm.^au),  is  to  be  evaluated  by  using  the  global  modes 
of  the  three-subsystem  model. 


3  Numerical  Examples  and  Variation  Analysis 

In  this  section,  two  examples  with  different  configurations  are  used  to  show  the  vari¬ 
ation  of  coupling  loss  factor  due  to  the  presence  of  a  third  coupled  subsystem.  The 
coupling  loss  factor  of  the  two-subsystem  model  is  evaluated  by  using  equations  (9) 
and  (10).  For  the  three-subsystem  model,  equations  (6)  and  (15)  are  used.  It  can  be 
shown  that  both  equations  (6)  and  (15)  give  the  same  results  as  the  couplings  are 
weak. 


3.1  Structural  details  and  SEA  model 


Ki 

beam  I  (TJT]  - 

(a)  ^  - Zi. 

beam  1  rnn  beam  2  (TTH  ^ 

(b)  ^  ^ - A  A - 


Figure  2;  A  structural  model  comprising  of  three  beams 

To  begin  example  calculations,  consider  initially  a  two-subsystem  model  (figure  2(a)) 
which  is  two  thin  beams  coupled  through  a  rotational  spring.  The  group  of  flexural 
vibration  modes  of  each  beam  are  taken  as  a  SEA  subsystem.  The  spring  provides 
weak  coupling  between  them  where  only  rotational  moment  is  transmitted.  When 
beam  3  is  connected  at  the  free  end  of  beam  2  to  the  original  two-beam  model, 
a  three-subsystem  model  is  formed  (figure  2(b)).  The  specifications  for  the  three 
beams  are  given  in  table  1.  The  spring  stiffness,  A'2,  is  adjustable  in  order  to  look 
into  the  significance  of  the  effect  of  the  third  subsystem.  There  are  two  cases  where 
the  length  of  beam  2  is:  (i)  L2  -  1.0?72;  (ii)  L2  =  l-lm.  The  spring  constants  at 
the  joints  are  chosen  to  be  weak  enough  to  ensure  that:(a)  the  coupling  loss  factor 
is  much  smaller  than  the  internal  loss  factor;  (b)  the  indirect  coupling  loss  factor  is 
much  smaller  than  the  direct  coupling  loss  factor. 

In  the  global  modal  approach  (see  section  2),  the  modes  of  two-subsystem  model 
and  three-subsystem  model  are  obtained  from  FEA.  In  numerical  simulation,  the 


1157 


Table  1 :  The  specifications  of  the  three  beams 


BEAM 

1 

2 

3 

length  (m) 

2.0 

1.0  &  1.1 

0.7 

width  (mm) 

4 

Thickness  (mm) 

2 

Density  (Kg/m^) 

7890 

Young’s  Modulus  (N/m'^) 

196E+9 

Poisson  Ratio 

0.29 

spring  constant,  A*i  (Nm/rad) 

1.0 

central  frequency  is  200Hz  and  the  averaging  band  is  selected  as  100  ~  300Hz.  In 
order  to  take  into  account  the  contribution  from  the  modes  out  of  the  band,  all  the 
modes  up  to  500Hz  are  extracted  for  evaluating  the  mobility  functions  in  averaging. 
The  modal  loss  factor  is  assumed  to  be  the  same  for  each  modes  used  in  averaging. 
This  means  that  the  internal  loss  factor  is  the  same  for  each  subsystem  and  is  equal 
to  the  modal  loss  factor  [11].  The  results  given  are  plotted  against  the  modal  loss 
factor  in  order  to  show  the  damping  effect  at  the  same  time. 


3.2  Results  and  discussion 

Figure  3  shows  the  identified  coupling  loss  factor  7/12  for  the  case  (i)  {Lo  =  1.0777) 
with  different  stiffness  of  /v'2.  The  case  of  /v'2  =  0  means  that  the  third  subsystem 


Figure  3:  7/12  is  negatively  affected  in  three-subsystem  model 
is  not  present.  It  can  be  seen  that  the  coupling  loss  factor  7^12  is  decreased  in  the  low 


1158 


range  of  damping  while  the  strength  of  the  coupling  between  subsystem  2  and  3  is 
increased.  The  stronger  the  coupling,  the  more  ijn  is  decreased. 

On  the  other  hand,  for  the  case  (ii)  where  L2  =  l.lm,  the  different  results  are 
shown  in  figure  4  where  the  presence  of  the  third  subsystem  would  mainly  increase 
i]i2  in  the  low  range  of  damping.  The  increasing  magnitude  is  also  dependent  on  the 
strength  of  coupling  between  subsystem  2  and  3.  The  explanation  for  the  different 
variation  trends  of  ?]i2  due  to  the  third  coupled  subsystem  between  figure  3  and  4 
will  be  given  later. 


Figure  4;  771  ■;  is  positively  affected  in  three-subsystem  model 

From  figure  3  and  4,  the  effect  of  damping  on  the  coupling  loss  factor  can  also  be 
observed.  In  the  low  damping  region,  increasing  damping  would  increase  coupling 
loss  factor.  After  a  certain  turnover  point,  increasing  damping  would  make  the 
coupling  loss  factor  decrease  and  finally  7712  becomes  convergent  to  a  value.  This 
agrees  with  the  conclusions  drawn  in  [10,  11].  It  is  shown  that,  even  though  the 
length  of  beam  2  has  a  slight  difference  in  figure  3  and  4,  the  converged  values  are 
still  very  close.  Thus,  the  converged  value  seems  not  to  depend  on  the  variation  of 
coupling  strength  at  A'2  and  the  structural  details,  although,  with  the  third  subsystem 
existing  in  the  system,  the  convergent  speed  is  faster.  Therefore,  it  is  reasonable  to 
believe  that  the  converged  coupling  loss  factor  at  sufficiently  high  damping  only 
depends  on  the  property  of  the  joint  rather  than  other  system  properties.  This  joint- 
dependent  property  of  coupling  loss  factor  in  the  high  range  of  damping  accords 
with  the  assumption  in  the  wave  approach.  Here,  it  is  convenient  to  define  the 
convergent  region  in  the  figure  3  and  4  as  the  “joint-dependent  zone”. 

However,  before  the  “joint-dependent  zone”,  coupling  loss  factor  seems  very 
sensitive  to  the  variation  of  damping  loss  factor  as  well  as  the  strength  of  coupling 
between  subsystem  2  and  3.  It  is  because  in  the  low  damping  region  the  system 


1159 


modal  properties  have  been  playing  a  major  role  in  determining  coupling  loss  fac¬ 
tor  [10, 11].  In  general,  the  coupling  loss  factor  represents  the  ability  of  energy 
transmitted  between  subsystems.  It  depends  not  only  on  the  physical  strength  of 
the  coupling  (e.g.,  spring  stiffness  in  the  examples),  but  also  on  the  amount  of  over¬ 
lap  between  the  modes  of  two  connected  subsystems.  The  higher  modal  overlap 
between  the  modes  of  two  connected  subsystem,  the  more  energy  is  transmitted 
between  the  subsystems.  As  a  result,  the  coupling  loss  factor  will  be  higher  even 
though  the  physical  strength  at  the  joint  is  unchanged.  If  the  modes  in  one  subsys¬ 
tem  are  distributed  exactly  the  same  as  those  in  one  another(for  instance,  two  exactly 
same  structures  are  coupled  together),  the  coupling  loss  factor  would  be  varied  to 
the  maximum,  and  vice  versa.  Therefore,  this  region  could  be  likely  defined  as 
“modal-sensitive  zone”. 

In  the  “modal-sensitive-zone”,  the  dependence  of  coupling  loss  factor  on  the 
amount  of  overlap  between  the  modes  of  two  connected  subsystems  has  been  clearly 
shown  in  figures  3  and  4.  For  the  case  (i)  illustrated  in  figure  3,  the  length  of  beam  2 
is  half  of  beam  1.  Due  to  the  characteristic  of  mode  distribution  in  beam  structure, 
the  amount  of  overlap  between  the  modes  of  subsystem  1  and  2  is  more  than  that  in 
the  case  (ii)  shown  in  figure  4,  where  beam  1  is  2  meters  and  beam  2  is  1.1  meters. 
Therefore,  case  (i)  has  higher  coupling  loss  factor  than  case  (ii)  in  “modal-sensitive 
zone”.  When  the  third  beam  is  coupled,  the  induced  variation  of  coupling  loss  factor 
depends  on  how  the  amount  of  overlap  between  the  modes  of  subsystems  1  and  2  is 
affected.  It  can  be  increased  or  decreased  and  thus  the  coupling  loss  factor  between 
subsystems  1  and  2  can  also  be  increased  or  decreased  due  to  the  third  coupled 
subsystem.  For  example,  the  amount  of  such  overlap  in  case  (i)  is  decreased  after 
the  third  subsystem  is  coupled.  As  a  result,  the  coupling  loss  factor,  7712,  becomes 
decreased. 

The  above  discussed  variability  of  coupling  loss  factor  due  to  the  third  coupled 
subsystem  has  been  shown  for  individual  cases.  On  the  other  hand,  if  an  ensemble 
of  similar  structures  are  considered,  this  sensitivity  may  be  reduced  (as  it  is  some¬ 
times  positive  or  negative  depending  on  each  special  situation).  However,  such  a 
variability  obtained  from  two  typical  examples  is  nevertheless  very  useful  when 
one  individual  case  is  studied  in  SEA  or  SEA-like  problems.  The  ignorance  of  such 
effect  of  the  other  coupled  subsystems  on  the  coupling  loss  factor  may  become  one 
of  the  possible  error  sources  causing  SEA  failure. 


4  Conclusions 

The  variation  of  coupling  loss  factor  due  to  the  third  coupled  subsystem  is  stud¬ 
ied  in  this  paper.  The  effect  of  a  third  coupled  subsystem  on  the  coupling  loss 
factor  between  the  first  two  coupled  subsystems  depends  on  how  strong  the  third 
subsystem  is  coupled.  Roughly,  along  with  the  damping  in  the  subsystems,  “joint- 
dependent  zone”  and  “modal-sensitive  zone”  are  defined  according  to  the  different 
variation  properties  of  coupling  loss  factor.  In  the  “modal-sensitive  zone”,  the  ef- 


1160 


feet  of  a  third  coupled  subsystem  on  the  coupling  loss  factor  could  be  positive  or 
negative.  It  depends  on  how  the  amount  of  overlap  between  the  modes  of  two  con¬ 
nected  subsystems  is  affected.  This  “modal-sensitive”  effect  may  be  averaged  out 
for  an  ensemble  of  structures,  but  it  is  important  when  SEA  is  applied  to  individual 
cases.  In  the  “joint-dependent  zone”,  the  coupling  loss  factor  is  insensitive  to  the 
strength  of  the  coupling  between  the  second  and  third  subsystems.  Since  the  two 
different  zones  are  allocated  according  to  the  system  damping  (which  is  equivalent 
to  modal  overlap  factor  when  the  central  frequency  and  modal  density  are  fixed), 
it  shows  the  importance  of  reverberance  in  subsystems  when  the  classical  SEA  is 
applied  [7,8].  How  to  take  into  account  the  effect  of  the  other  coupled  subsystems 
in  evaluating  coupling  loss  factor,  especially  when  the  system  damping  is  low  and 
when  an  individual  case  is  considered,  definitely  needs  to  be  further  investigated. 


References 

[  1  ]  Richard  H.  Lyon  and  Richard  G.  DeJong.  Theory  and  Application  of  Statistical 
Energy  Analysis.  Butterworth-Heinemann,  second  edition,  1995. 

[2]  Richard  H.  Lyon  and  G.  Maidanik.  Power  flow  between  linearly  coupled  os¬ 
cillators.  Journal  of  the  Acoustic  Society  of  America,  34:623-639,  1962. 

[3]  Eric  E.  Ungar.  Statistical  energy  analysis  of  vibrating  systems.  Transactions 
of  the  ASME,  Journal  of  Engineering  for  Industry,  pages  626-632,  November 
1967. 

[4]  F.  J.  Fahy.  Statistical  energy  analysis.  In  R.  G.  White  and  J.  G.  Walker,  editors. 
Noise  and  Vibration,  chapter  7,  pages  165-186.  Chichester,  Ellis  Horwood, 
1982. 

[5]  J.  M.  Cuschieri  and  J.  C.  Sun.  Use  of  statistical  energy  analysis  for  rotating 
machinery,  part  II;  Coupling  loss  factors  between  indirectly  coupled  substruc¬ 
tures.  Journal  of  Sound  and  Vibration,  170(2):  191-201,  1994. 

[6]  P.  W.  Smith.  Statistical  models  of  coupled  dynamical  systems  and  the  transi¬ 
tion  from  weak  to  strong  coupling.  Journal  of  the  Acoustic  Society  of  America, 
65:695-698,  1979. 

[7]  S.  Finnveden.  Ensemble  averaged  vibration  energy  flows  in  a  three-element 
structure.  Journal  of  Sound  and  Vibration,  187(3);495-529,  1995. 

[8]  A.  J.  Keane.  A  note  on  modal  summations  and  averaging  methods  as  ap¬ 
plied  to  statistical  energy  analysis  (SEA).  Journal  of  Sound  and  Vibration, 
164(1);  143-156,  1993. 

[9]  B.  R.  Mace.  The  statistical  energy  analysis  of  two  continuous  one-dimensional 
subsystems.  Journal  of  Sound  and  Vibration,  166(3):429-461,  1993. 


1161 


[  10]  Hongbing  Du  and  Fook  Fah  Yap.  A  study  of  damping  effects  on  coupling  loss 
factor  used  in  statistical  energy  analysis.  In  Proceedings  of  the  Fourth  Inter¬ 
national  Congress  on  Sound  and  Vibration,  pages  265-272,  St.  Petersburge, 
Russia,  June  1996. 

[11]  Fook  Fah  Yap  and  J.  Woodhouse.  Investigation  of  damping  effects  on  statis¬ 
tical  energy  analysis  of  coupled  structures.  Journal  of  Sound  and  Vibration, 
197(3):35I-371,  1996. 

[12]  D.  A.  Bies  and  S.  Hamid.  In  situ  determination  of  loss  and  coupling  loss 
factors  by  the  power  injection  method.  Journal  of  Sound  and  Vibration, 
70(2):  187-204,  1980. 

[13]  L.  Cremer,  M.  Heckl,  et  al.  Structure-Borne  Sound:  Structural  Vibrations  and 
Sound  Radiation  at  Audio  Frequencies.  Springer- Verlag,  second  edition,  1987. 

[14]  R.  S.  Langley.  A  derivation  of  the  coupling  loss  factors  used  in  statistical 
energy  analysis.  Journal  of  Sound  and  Vibration,  141(2):207-219,  1990. 

[15]  J.  C.  Sun,  C.  Wang,  et  ai.  Power  flow  between  three  series  coupled  oscillators. 
Journal  of  Sound  and  Vibration,  1 89(2)  :2 15-229,  1996. 


1162 


THE  EFFECT  OF  CURVATURE  UPON  VIBRATIONAL 
POWER  TRANSMISSION  IN  BEAMS 
SJ.  Walsh(l)  and  R.G.White(2) 


(1)  Department  of  Aeronautical  and  Automotive 
Engineering  and  Transport  Studies 
Loughborough  University 

(2)  Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 

ABSTRACT 

Previous  research  into  structural  vibration  transmission  paths 
has  shown  that  it  is  possible  to  predict  vibrational  power 
transmission  in  simple  beam  and  plate  structures.  However,  in 
many  practical  structures  transmission  paths  are  composed  of  more 
complex  curved  elements;  therefore,  there  is  a  need  to  extend 
vibrational  power  transmission  analyses  to  this  class  of  structure. 
In  this  paper,  expressions  are  derived  which  describe  the  vibrational 
power  transmission  due  to  flexural,  extensional  and  shear  types  of 
travelling  wave  in  a  curved  beam  which  has  a  constant  radius  of 
curvature.  By  assuming  sinusoidal  wave  motion,  expressions  are 
developed  which  relate  the  time-averaged  power  transmission  to 
the  travelling  wave  amplitudes.  The  results  of  numerical  studies 
are  presented  which  show  the  effect  upon  power  transmission 
along  a  curved  beam  of:  (i)  the  degree  of  curvature;  and  (ii)  various 
simplifying  assumptions  made  about  the  beam  deformation. 

1.  INTRODUCTION 

Previous  research  into  structural  transmission  paths  has 
shown  it  is  possible  to  predict  vibrational  power  transmission  in 
simple  beam  and  plate  structures.  More  recently,  transmission 
through  pipes  with  bends,  branches  and  discontinuities  has  been 
studied,  which  has  led  to  useful  design  rules  concerning  the 
position  and  size  of  pipe  supports  for  minimum  power 
transmission[l].  However,  in  many  practical  structures 
transmission  paths  are  composed  of  more  complex  curved 
elements.  Therefore,  there  is  a  need  to  extend  power  transmission 
analyses  to  this  class  of  structure. 


1163 


Wave  motion  in  a  curved  beam  with  a  constant  radius  of 
curvature  has  been  considered  by  Love  [2]  who  assumed  that  the 
centre-line  remains  unextended  during  flexural  motion,  whilst 
flexural  behaviour  is  ignored  when  considering  extensional 
motion.  Using  these  assumptions  the  vibrational  behaviour  of 
complete  or  incomplete  rings  has  been  considered  by  many 
researchers  who  are  interested  in  the  low  frequency  behaviour  of 
arches  and  reinforcing  rings.  In  reference  [2]  Love  also  presented 
equations  for  thin  shells  which  include  the  effects  of  extension  of 
the  mid-surface  during  bending  motion.  Soedel  [3]  reduced  these 
equations  and  made  them  applicable  to  a  curved  beam  of  constant 
radius  of  curvature.  In  an  alternative  approach  Graff  [4]  derived 
these  equations  from  first  principles  and  also  constructed  frequency 
verses  wavenumber  and  wavespeed  versus  wavenumber  graphs. 
Philipson  [5]  derived  a  set  of  equations  of  motion  which  included 
extension  of  the  central  line  in  the  flexural  wave  motion,  and  also 
rotary  inertia  effects.  In  a  development  analogous  to  that  of 
Timoshenko  for  straight  beams,  Morley  [6]  introduced  a  correction 
for  radial  shear  when  considering  the  vibration  of  curved  beams. 
Graff  later  presented  frequency  versus  wave  number  and  wave 
speed  versus  wave  number  data  for  wave  motion  in  a  curved  beam, 
when  higher  order  effects  are  included  [7]. 

In  this  paper,  expressions  for  vibrational  power  transmission 
in  a  curved  beam  are  derived  from  first  principles.  In  the  next 
section  two  sets  of  governing  equations  for  wave  motion  in  a 
curved  beam  are  presented  both  of  which  include  coupled 
extensional-flexural  motion.  The  first  set  is  based  upon  a  reduction 
of  Love's  thin  shell  equations  mentioned  above.  The  second  set  is 
based  upon  a  reduction  of  Fliigge’s  thin  shell  equations  [8].  In 
section  three,  the  expressions  for  stresses  and  displacements 
presented  in  section  two  are  used  to  derive  formulae  for  vibrational 
power  transmission  in  terms  of  centre-line  displacements.  By 
assuming  sinusoidal  wave  motion,  expressions  are  developed 
which  relate  the  time-averaged  power  transmission  to  the 
extensional  and  flexural  travelling  wave  amplitudes.  In  section 
four,  corrections  for  rotary  inertia  and  shear  deformation  are 
introduced.  The  results  of  numerical  studies  of  these  expressions 
are  presented  which  show  the  effect  upon  wave  motion  and  power 
transmission  of  (i)  the  degree  of  curvature,  and  (ii)  the  various 
simplifying  assumptions  made  about  the  beam  deformation. 


1164 


2.  WAVE  MOTION  IN  CURVED  BEAMS 


In  this  section  the  governing  relations  between  displacements, 
strains,  stresses  and  force  resultants  in  a  curved  beam  are  presented. 
The  centre-line  of  the  beam  lies  in  a  plane  and  forms  a  constant 
radius  of  curvature.  The  cross-section  of  the  beam  is  uniform  and 
symmetrical  about  the  plane  and  it  is  assumed  that  there  is  no 
motion  perpendicular  to  the  plane.  It  is  also  assumed  that  the  beam 
material  is  linearly  elastic,  homogeneous,  isotropic  and  continuous. 

Consider  a  portion  of  the  curved  beam,  as  shown  in  Figure  1. 
The  circumferential  coordinate  measured  around  the  centre-line  is 
s,  while  the  outward  pointing  normal  coordinate  from  the  centre¬ 
line  is  z,  and  the  general  radial  coordinate  is  r.  A  complete  list  of 
notation  is  given  in  the  appendix.  For  small  displacements  of  thin 
beams  the  assumptions,  known  as  "Love's  first  approximation"  in 
classical  shell  theory,  can  be  made  [8].  This  imposes  the  following 
linear  relationships  between  the  tangential  and  radial 
displacements  of  a  material  point  and  components  of  displacement 
at  the  undeformed  centre-line: 


U  (r,  s,  t)  =  u  (R,  s,  t)  +  z  (|)  (s,  t)  (1) 

W  (r,  s,  t)  =  w  (R,  s,  t)  (2) 

where  u  and  w  are  the  components  of  displacement  at  the  centre¬ 
line  in  the  tangential  and  radial  directions,  respectively,  (j)  is  the 
rotation  of  the  normal  to  the  centre-line  during  deformation: 


(|> 


3w 


/  angle  of'X  { rotational  displacement^ 

Vcurvature/  \  of  straight  beam  ) 


(3) 


and  W  is  independent  of  z  and  is  completely  defined  by  the  centre¬ 
line  component  w. 

Circumferential  strain  consists  of  both  an  extensional  strain 
and  bending  strain  component.  Expressions  for  these  are  listed  in 
table  1.  The  strain-displacement  expressions  of  the  Love  and  Flugge 
based  equations  are  identical.  However,  in  the  total  circumferential 
strain  of  the  Love  based  theory,  the  term  in  the  denominator 
has  been  neglected  with  respect  to  unity.  Assuming  the  material  to 
be  linearly  elastic,  the  circumferential  stress-strain  relationship 
is  given  by  Hooke's  Law,  whilst  the  shear  strain,  Ysr/  and  shear 
stress,  Osr/  are  assumed  to  be  zero.  Assuming  the  material 


1165 


to  be  homogeneous  and  isotropic,  the  material  properties  E,  G  and  v 
can  be  treated  as  constants.  Thus,  by  integrating  the  stresses  over 
the  beam  thickness,  force  and  moment  resultants  can  be  obtained, 
which  are  listed  in  table  2.  The  adopted  sign  convention  is  shown  in 
Figure  2. 

Equations  of  motion  for  a  curved  beam  are  presented  in  [4]. 
These  equations  are  derived  in  terms  of  the  radian  parameter  0.  By 
applying  the  substitution,  s  -  RQ,  the  equations  of  motion  can  be 
expressed  in  terms  of  the  circumferential  length,  s.  These  equations 
are  listed  in  [9]  along  with  the  Fliigge  based  equations  of  motion 
which  have  been  obtained  by  a  reduction  of  the  equations  of 
motion  for  a  circular  cylindrical  shell  presented  in  [8].  An  harmonic 
solution  of  the  equations  of  motion  can  be  obtained  by  assuming 
that  extensional  and  flexural  sinusoidal  waves  propagate  in  the 
circumferential  direction.  The  harmonic  form  of  the  equations  of 
motion  are  also  listed  in  [9]. 


3.  VIBRATIONAL  POWER  TRANSMISSION  IN  CURVED  BEAMS 

In  this  section  the  expressions  for  displacements  and  stresses 
presented  in  section  two  are  used  to  derive  the  structural  intensity 
and  power  transmission  due  to  flexural  and  extensional  travelling 
waves  in  a  curved  beam.  The  structural  intensity  expressions  are 
formulated  in  terms  of  displacements  at  the  centre-line.  By 
assuming  sinusoidal  wave  motion,  expressions  are  developed 
which  relate  the  time-averaged  power  transmission  to  the  flexural 
and  extensional  travelling  wave  amplitudes. 

Structural  Intensity  in  the  circumferential  direction  of  a  curved 
beam  is  given  by  [10]: 


Is  = 


au 
at 

f  intensity  due  to 
I  circumferential  stress 


aw 
at 

f  intensity  due  to 
radial  shear  stress 


(4) 


By  integrating  across  the  beam  thickness  power  transmission  per 
unit  length  in  the  circumferential  direction  is  obtained: 

h/2 

Ps=  J  Isdz  (5) 

-h/2 


1166 


Substituting  the  Love  based  circumferential  stress-strain  relation 
and  strain-displacement  expression  into  equation  (5)  the  power 
transmission  due  to  circumferential  stress  is  obtained.  (A  full 
derivation  is  given  in  [9].)  By  analogy  to  power  transmission  in  a 
straight  beam  [1]  this  can  be  expressed  in  terms  of  an  extensional 
component,  Pe,  and  a  bending  moment  component,  Pbm*  Although 
the  transverse  shear  stress  Ogr  is  negligible  under  Love's  first 
approximation,  the  power  transmission  due  to  transverse  shear 
stress  can  be  evaluated  from  the  non-vanishing  shear  force,  Q, 
because  the  radial  displacement  W  does  not  vary  across  the  beam 
thickness.  Again,  by  analogy  to  power  transmission  in  a  straight 
beam  [1]  this  is  expressed  as  a  shear  force  component.  Thus,  the  total 
power  transmission  in  the  circumferential  direction  is  given  by  the 
sum  of  the  extensional,  bending  moment  and  shear  force 
components.  These  equations  are  listed  in  table  3  along  with  Fliigge 
based  power  transmission  equations  which  are  also  listed  in  table  3. 
Substituting  harmonic  wave  expressions  into  the  Love  and  Fliigge 
based  power  transmission  equations  gives  expressions  for  the  power 
transmission  in  the  circumferential  direction  in  terms  of  travelling 
wave  amplitudes  A  and  B.  For  sinusoidal  wave  motion  it  is  useful 
to  develop  time-averaged  power  transmission  defined  by  [1]: 


T/2 


<Ps> 


--  f 

t“T  J 


Ps  (s,t)  dt 


-  T/2 


(6) 


where  T  is  the  period  of  the  signal.  Time  averaged  Love  and  Fliigge 
based  power  transmission  equations  are  given  in  table  4. 


4.  THE  EFFECT  OF  ROTARY  INERTIA  AND  SHEAR 
DEFORMATION 

It  is  known  that  shear  deformation  and  rotary  inertia  effects 
become  significant  for  straight  beams  as  the  wave  length  approaches 
the  same  size  as  the  thickness  of  the  beam,  and  for  cylindrical  shells 
as  the  shell  radius  decreases  [8].  Thus,  the  objective  in  this  section  is 
to  establish  more  complete  equations  for  power  transmission  in  a 
curved  beam  and  to  show  under  what  conditions  these  specialise  to 
the  simple  bending  equations  presented  in  section  three. 

Rotary  inertia  effects  are  included  by  considering  each  element 
of  the  beam  to  have  rotary  inertia  in  addition  to  translational 
inertia.  Equations  of  motion  for  a  curved  beam  which  include  the 


1167 


effect  of  rotary  inertia  are  presented  in  [7].  These  equations  are 
listed  in  [9]  in  terms  of  the  circumferential  distance  parameter,  s. 
Equations  for  vibrational  power  transmission  can  be  derived  in  the 
same  marmer  as  described  in  section  three.  These  equations  are 
listed  in  tables  3  and  4  where  it  can  be  seen  that  the  extensional  and 
bending  moment  components  when  including  rotary  inertia  effects 
are  identical  to  the  corresponding  Fliigge  based  expressions. 
However,  the  shear  force  component  now  contains  an  additional 
rotary  inertia  term. 

If  shear  deformation  is  included  then  Kirchoff  s  hypothesis  is 
no  longer  valid,  and  the  rotation  of  the  normal  to  the  centre-line 
during  bending,  (|),  is  no  longer  defined  by  equation  (3)  but  is  now 
another  independent  variable  related  to  the  shear  angle,  y. 
However,  unlike  simple  bending  theory,  where  the  transverse 
shear  strain,  Ysr^  is  negligible,  the  transverse  shear  strain  is  now 
related  to  the  shear  angle,  y  which  is  expressed  in  terms  of 
displacements  u,  w  and  (j).  The  circumferential  force,  bending 
moment,  and  shear  force  obtained  from  [8]  are  given  in  table  2.  A  set 
of  equations  of  motion  for  a  curved  beam  which  includes  the  effect 
of  shear  deformation  is  presented  in  [9].  Power  transmission 
equations  in  the  circumferential  direction  can  be  obtained  in  a 
manner  analogous  to  that  used  for  Love  and  Fliigge  based  theories. 
As  before,  the  power  transmission  due  to  circumferential  stress 
can  be  identified  as  consisting  of  extensional  and  bending  moment 
components.  The  contribution  to  the  power  transmission  from  the 
transverse  shear  stress  is  obtained  from  the  product  of  the  shear 
force  resultant  and  the  radial  velocity  which  gives  the  shear  force 
component  of  power  transmission. 


5.  NUMERICAL  STUDY 

For  a  given  real  wavenumber,  k,  the  harmonic  equations  of 
motion  were  solved  to  find  the  corresponding  circular  frequency,  co, 
and  complex  wave  amplitude  ratio.  The  simulated  beam  was 
chosen  to  have  the  physical  dimensions  and  material  properties  of 
typical  mild  steel  beams  used  for  laboratory  experiments.  Four 
different  radii  of  curvature  were  investigated,  which  were 
represented  in  terms  of  the  non-dimensional  thickness  to  radius  of 
curvature  ratio,  h/R.  These  ratios  were  ^/lO/ 100/ 1000  and 
^/lOOOO- 

Using  the  Love  equations  of  motion.  Figure  3  shows  the 
relationship  between  wave  number  and  frequency  for  a  beam  with  a 


1168 


thickness  to  radius  of  curvature  ratio  of  ^/lo-  The  frequency  range 
is  represented  in  terms  of  the  non-dimensional  frequency 
parameter  Q  =  ®^/co,  where  Cq  is  the  phase  velocity  of  extensional 
waves  in  a  straight  bar  and  the  wave  number  range  is  represented 
in  terms  of  the  non-dimensional  wave  number,  kR.  It  can  be  seen 
that  two  types  of  elastic  wave  exist:  one  involving  predominantly 
flexural  motion;  the  other  predominantly  extensional  motion. 
However,  for  wave  numbers  less  than  kR  =  1,  the  predominantly 
flexural  wave  exhibits  greater  extensional  than  flexural  motion. 

Solution  of  the  shear  deformation  equations  of  motion  for  a 
curved  beam  shows  that  three  types  of  elastic  wave  exist.  These  are 
the  predominantly  flexural  and  predominantly  extensional  waves 
of  simple  bending  theory  and  additionally  a  predominantly 
rotational  wave  related  to  the  shear  angle.  The  relationship 
between  wave  number  and  frequency  for  these  three  wave  types  is 
shown  in  Figure  4. 

A  numerical  investigation  of  the  power  transmission 
equations  was  undertaken  using  simulated  beams  with  the  same 
dimensions  and  material  properties  as  those  used  in  the  previous 
study  of  wave  motion.  Figure  5  shows  the  relationship  between 
transmitted  power  ratio  and  frequency.  For  the  predominantly 
flexural  wave  the  time-averaged  transmitted  power  ratio  is 
calculated  by  dividing  the  time-averaged  power  transmitted  along  a 
curved  beam  by  a  predominantly  flexural  wave  by  the  time- 
averaged  power  transmitted  by  a  pure  flexural  wave  travelling  in  a 
straight  Euler-Bernoulli  beam.  i.e.  the  ratio  (<Pe>t  +  <Fbm>t  + 
<Psf>t)/EIcokf3Af .  For  the  predominantly  extensional  wave  the 
transmitted  power  ratio  is  calculated  by  dividing  the  time-averaged 
power  transmitted  along  a  curved  beam  by  a  predominantly 
extensional  wave  by  the  time-averaged  power  transmitted  by  a  pure 
extensional  wave  in  a  straight  rod.  i.e.  the  ratio  (<Pe>t  +  <Pbn>t  + 
<Psf  >t)/EScokex  P^ex 

6.  SUMMARY  AND  CONCLUSIONS 

In  this  paper,  starting  from  first  principles,  expressions  for 
vibrational  power  transmission  in  a  curved  beam  have  been 
derived  using  four  different  theories.  Love’s  generalised  shell 
equations  include  extension  of  the  centre-line  during  bending 
motion  were  the  first  set  of  equations  considered.  Fiiigge's 
equations  also  include  centre-line  extensions  and  were  the  second 
set  of  equations  used.  Corrections  for  rotary  inertia  and  shear 
deformation  produced  the  third  and  fourth  sets  of  governing 


1169 


equations,  respectively.  By  letting  the  radius  of  curvature,  R,  tend 
to  infinity  these  equations  reduce  to  the  corresponding  straight 
beam  expressions  presented  in  [1]. 

Using  the  governing  equations  for  each  theory,  expressions 
were  then  developed  which  related  time-averaged  power 
transmission  to  the  amplitudes  of  the  extensional,  flexural  and 
rotational  displacements.  For  each  theory  the  effects  of  curvature 
upon  the  resulting  wave  motion  and  power  transmission  were 
then  investigated  using  beams  with  different  degrees  of  curvature. 
From  the  results  of  this  study  it  can  be  seen  that  vibrational  power 
transmission  in  curved  beams  can  be  classified  into  three  different 
frequency  regions: 

(i)  below  the  ring  frequency,  Q  =  1,  curvature  effects  are 
important; 

(ii)  above  the  ring  frequency  but  below  the  shear  wave  cut-on 

frequency,  =  1  the  curved  beam  behaves  essentially  as  a 

straight  beam; 

(iii)  above  the  shear  wave  cut-on  frequency,  higher  order  effects 
are  important. 


ACKNOWLEDGEMENT 

The  analytical  work  presented  in  this  paper  was  carried  out  while 

both  authors  were  at  the  Institute  of  Sound  and  Vibration  Research, 

University  of  Southampton.  The  financial  support  of  the  Marine 

Technology  Directorate  Limited  and  the  Science  and  Engineering 

Research  Council  is  gratefully  acknowledged. 

REFERENCES 

1.  J.L.  HORNER  1990  PhD  thesis,  University  of  Southampton 
Vibrational  power  transmission  through  beam  like  structures. 

2.  A.E.H.  LOVE  1940  Dover,  Nezv-York.  A  treatise  on  the 
mathematical  theory  of  elasticity. 

3.  W.  SOEDEL  1985  Dekker,  New  York.  Vibrations  of  shells  and 
plates 

4.  K.F.  GRAFF  1975  Clarenden  Press,  Oxford.  Wave  motion  in 
elastic  solids 

5.  L.L.  PHILIPSON  1956  Journal  of  Applied  Mechanics  23,  364- 
366.  On  the  role  of  extension  in  the  flexural  vibrations  of  rings. 

6.  L.S.D.  MORLEY  1961  Quarterly  Journal  of  Mechanics  and 
Applied  Mathematics  14,  (2),  155-172.  Elastic  waves  in  a 
naturally  curved  rod. 


1170 


7.  K.F.  GRAFF  1970  IEEE  Transactions  on  Sonics  and  Ultrasonics, 
SU-17  (1),  1-6.  Elastic  wave  propagation  in  a  curved  sonic 
transmission  line. 

8.  A.W.  LEISSA  1977  NASA  SP-288,  Washington  DC  Vibrations 
of  shells. 

9.  S.J.  WALSH  1996  PhD  thesis,  University  of  Southampton. 
Vibrational  power  transmission  in  curved  and  stiffened 
structures. 

10.  A.J.  ROMANO,  P.B.  ABRAHAM,  E.G.  WILLIAMS  1990  Journal 
of  the  Acoustical  Society  of  America  87.  A  Poynting  vector 
formulation  for  thin  shells  and  plates,  its  application  to 
structural  intensity  analysis  and  source  localization.  Part  I: 
Theory. 


APPENDIX:  NOTATION 

A  flexural  wave  amplitude 

Af  amplitude  of  a  purely  flexural  wave 

B  extensional  wave  amplitude 

Bex  amplitude  of  a  purely  extensional  wave 

C  rotation  wave  amplitude 

E  Young’s  modulus 

G  shear  modulus 

I  second  moment  of  area  of  cross-section  of  beam 

Ig  structural  intensity  in  circumferential  direction 

K  radius  of  gyration 

M  bending  moment  on  cross-section  of  beam 

N  circumferential  force  on  cross-section  of  beam 

P  transmitted  power 

Q  shear  force  on  cross-section  of  beam 

R  radius  of  curvature 

S  cross-sectional  area  of  beam 

T  period  of  wave 

LF  displacement  in  circumferential  direction 

W  displacement  in  radial  direction 

Co  wavespeed  of  extensional  waves  in  a  straight  bar 

Cs  wavespeed  of  shear  waves  in  a  straight  bar 

ds  length  of  elemental  slice  of  curved  beam 

eg  total  circumferential  strain 

h  thickness  of  beam 

k  wavenumber 

kex  wavenumber  of  a  purely  extensional  wave 

kf  wavenumber  of  a  purely  flexural  wave 

r  coordinate  in  radial  direction 


1171 


s  coordinate  in  circumferential  direction 

u  displacement  at  centre-line  in  circumferential  direction 

w  displacement  at  centre-line  in  radial  direction 

z  coordinate  of  outward  pointing  normal 

Q.  non-dimensional  frequency 

Ps  bending  strain 

Y  shear  angle 

Ysr  transverse  shear  strain 

Ej.  radial  strain 

£g  circumferential  strain 

0A  phase  angle  of  flexural  wave 

0C  phase  angle  of  rotational  wave 

K  Timoshenko  shear  coefficient 

?iex  wave  length  of  extensional  waves  in  a  straight  bar 

V  Poisson's  ratio 

Oj.  radial  stress 

Gg  circumferential  stress 

0SJ  transverse  shear  stress 

d  change  in  slope  of  normal  to  centre-line 

CO  radian  frequency 


W(r,s,t) 


1172 


Figure  2:  Sign  convention  and  force  resultants  on  an  elemental 
slice  of  curved  beam 


Figure  3:  Wave  number  v.  frequency  relationship  for  a  curved 
beam  predicted  using  Love  theory 


Transmitted  power  ratio  (Curved  beam)/(Slraight  beam)  ^  Non-dimensional  wavenumber 


Table  1:  Displacement,  strain-displacement  and  stress-strain  equations  for  a  curved  beam. 


1175 


Table  2:  Force  resultants  for  a  curved  beam. 


1176 


Table  3:  Power  transmission  for  a  curved  beam. 


1177 


Table  4:  Time-averaged  power  transmission  by  a  single  harmonic  wave 


1178 


A  Parameter-based  Statistical  Energy 
Method  for  Mid-frequency 
Vibration  Transmission  Analysis 

Sungbae  Choi,  Graduate  Student  Research  Assistant 
Matthew  P.  Castanier,  Assistant  Research  Scientist 
Christophe  Pierre,  Associate  Professor 
Department  of  Mechanical  Engineering  and  Applied  Mechanics 
The  University  of  Michigan 
Ann  Arbor,  MI  48109-2125,  USA 

Abstract 

Vibration  transmission  between  two  multi-mode  substructures  con¬ 
nected  by  a  spring  is  investigated.  A  classical  Statistical  Energy 
Analysis  (SEA)  approach  is  reviewed,  and  it  is  seen  that  some  typ¬ 
ical  assumptions  which  are  valid  at  high  frequencies  lose  accuracy 
in  the  mid-frequency  range.  One  assumption  considered  here  is 
that  of  an  identical  probability  density  function  (pdf)  for  each  reso¬ 
nant  frequency.  This  study  proposes  a  Parameter-based  Statistical 
Energy  Method  (PSEM)  which  considers  individual  modal  informa¬ 
tion.  The  results  of  PSEM  have  good  agreement  with  those  of  a 
Monte  Carlo  technique  for  an  example  system. 

Nomenclature 


E[  ] 

expected  value 

power  transmitted  between  substructure  i  and  k 

IIfc(u;)  . 

total  power  transmitted  to  substructure  k 

Pi 

power  input  to  substructure  i 

Vi, 

coupling  loss  factor  (CLF) 

fh 

modal  driving  force  for  mode  j  of  substructure  1 

UJ 

frequency  [rad/sec] 

l.l.-.lj.Ol 

subscripts  for  decoupled  Bar  1 

2,2r,2„02 

subscripts  for  decoupled  Bar  2 

El,  ^2 

blocked  energy 

Eoi ,  Eo2 

Young’s  modulus 

Pl^P2 

density 

mi, m2 

mass  per  unit  length 

Mu  M2 

total  mass 

Cl,C2 

viscous  damping  ratio 

Ai,  A2 

cross-sectional  area 

0 

rH 

0 

nominal  length 

1179 


Ti,T2 

disordered  length 

£1,^2 

ratio  of  disorder  to  nominal  length 

k 

coupling  stiffness 

coupling  ratio 

Xi,X2 

position  coordinate 

ai,a2 

point- coupling  connection  position 

Wi,  W2 

deflection 

modal  amplitude 

mode  shape  function 

NuN2 

number  of  resonant  frequencies 

resonant  frequencies 

lower  limit  of  resonant  frequencies 

upper  limit  of  resonant  frequencies 

(Tl,  (72 

standard  deviation  of  disorder 

1  Introduction 

Vibration  transmission  analysis  between  connected  substructures  in  the 
mid-frequency  range  is  often  a  daunting  prospect.  Since  the  analysis  at 
high  frequencies  requires  greater  model  discretization,  the  size  and  com¬ 
putational  cost  of  a  full  structure  model  (e.g.,  a  Finite  Element  model) 
can  become  prohibitive.  Also,  as  the  wavelengths  approach  the  scale  of 
the  structural  variations,  uncertainties  (tolerances,  defects,  etc.)  can  sig¬ 
nificantly  affect  the  dynamics  of  the  structure.  Starting  at  what  may  be 
called  the  mid-frequency  range,  deterministic  models  fail  to  predict  the 
response  of  a  representative  structure  with  uncertainties. 

Therefore,  in  the  mid-frequency  range,  a  statistical  analysis  of  vibra¬ 
tion  transmission  may  be  more  appropriate.  This  approach  is  taken  in 
the  procedure  known  as  Statistical  Energy  Analysis  (SEA)  [1].  In  SEA, 
a  structure  is  divided  into  coupled  substructures.  It  is  assumed  that  each 
substructure  exhibits  strong  modal  overlap  which  makes  it  difficult  to  dis¬ 
tinguish  individual  resonances.  Therefore,  the  resonant  frequencies  are 
treated  as  random  variables,  each  with  an  identical,  uniform  probability 
density  function  (pdf)  in  the  frequency  range  of  interest.  This  assumption 
greatly  simplifies  the  evaluation  of  the  expected  value  of  transmitted  vi¬ 
bration  energy.  A  simple  linear  relation  of  vibration  transmission  between 
each  pair  of  substructures  is  retrieved.  The  power  transmitted  is  propor¬ 
tional  to  the  difference  in  the  average  modal  energies  of  the  substructures. 
This  relation  is  analogous  to  Fourier’s  law  of  heat  transfer  [1-4]. 

In  the  low-  to  mid-frequency  region,  the  modal  responses  are  not  strongly 
overlapped.  In  this  case,  two  typical  SEA  assumptions  are  less  accurate:  an 
identical  pdf  for  all  resonant  frequencies,  and  identical  (ensemble-averaged) 


1180 


values  of  the  associated  mode  shape  functions  at  connection  positions.  In 
this  paper,  these  two  assumptions  are  relaxed.  A  distinct  uniform  pdf 
is  applied  for  each  resonant  frequency,  and  a  piecewise  evaluation  of  the 
transmitted  power  is  performed.  This  is  called  a  Parameter-based  Sta¬ 
tistical  Energy  Method  (PSEM)  because  it  considers  the  statistical  char¬ 
acteristics  of  individual  system  parameters.  This  solution  can  accurately 
capture  peaks  of  transmitted  power  while  maintaining  the  SEA  advantage 
of  efficiency. 

This  paper  is  organized  as  follows.  In  section  2,  we  briefly  review  SEA 
along  with  the  associated  assumptions  and  limitations.  In  section  3,  the 
power  transmitted  between  two  spring- coupled  multi-mode  substructures 
is  investigated  by  applying  several  SEA  assumptions.  A  Monte  Carlo  solu¬ 
tion  is  used  for  comparison.  In  section  4,  the  PSEM  approach  is  presented 
and  the  results  are  shown.  Finally,  section  5  draws  conclusions  from  this 
study. 

2  Overview  of  SEA 

In  Statistical  Energy  Analysis,  the  primary  variable  is  the  time-averaged 
total  energy  of  each  substructure.  This  is  called  the  blocked  energy,  where 
blocked  means  an  assumed  coupling  condition.  The  assumed  coupling  con¬ 
dition  may  be  the  actual  coupling,  a  clamped  condition  at  the  substructure 
junctions,  or  a  decoupled  condition  [2,  5]. 

In  order  to  predict  the  average  power  transmitted  between  two  directly- 
coupled  substructures,  a  few  simplifying  assumptions  are  applied.  Some  of 
the  essential  SEA  assumptions  are  summarized  by  Hodges  and  Woodhouse 
in  Ref.  [3]; 

•  Modal  incoherence:  the  responses  of  two  different  modal  coordinates 
are  uncorrelated  over  a  long  time  interval 

•  Equipartition  of  modal  energy:  all  modes  within  the  system  have  the 
same  kinetic  energy 

The  above  conditions  make  it  possible  to  treat  all  modal  responses  as  sta¬ 
tistically  identical.  The  first  assumption  implies  a  broad  band,  distributed 
driving  force  (often  called  ”rain  on  the  roof’)  which  leads  to  uncorrelated 
modal  driving  forces.  The  second  assumption  implies  that  the  substruc¬ 
tures  have  strong  modal  overlap,  or  that  the  parameter  uncertainties  are 
sufficiently  large  that  the  modes  are  equally  excited  in  an  ensemble  average 
sense.  Thus,  the  resonant  frequencies  are  treated  as  random  variables  with 
identical,  uniform  probability  density  functions  (pdfs)  for  the  frequency 
range  of  interest. 

The  SEA  relation  for  the  expected  value  of  power  transmitted  from 


1181 


substructure  i  to  substructure  k,  may  be  expressed  as 

E[nijfc(cj)]  =  0)7}, ^Ni{^  (1) 

where  uj  is  the  frequency,  77^^  is  the  coupling  loss  factor,  Ei  is  the  blocked 
energy  of  substructure  z,  and  Ni  is  the  number  of  participating  modes  of 
substructure  i  for  the  frequency  range  of  interest.  The  power  dissipated  by 
substructure  i  is  expressed  as 

=  u:riiEi  (2) 

where  r]i  is  the  damping  loss  factor.  Using  Eqs.(l)  and  (2),  the  equation 
of  power  balance  for  substructure  i  at  steady  state  [1,  2,  5,  6]  is 

N  N 

Pi  =  E[Ilij]  +  UJTJiEi  =  UJ  (3) 

i=i 

j¥i 

where  Pi  is  the  power  input  to  substructure  i  from  external  sources.  Note 
that  the  first  term  on  the  right-hand  side  is  the  power  transmitted  through 
direct  coupling  between  substructures. 


3  Vibration  Transmission  in  a  Two-Bar 

System 

The  longitudinal  vibration  of  the  structure  shown  in  Fig.l  is  considered 
in  this  study.  The  structure  consists  of  two  uniform  bars  with  viscous 
damping  which  are  coupled  by  a  linear  spring  of  stiffness  k.  The  spring  is 
connected  at  intermediate  points  on  the  bars,  Xi  =  ai  and  X2  =  0,2.  Bar 
i  has  nominal  length  Loi.  A  parameter  uncertainty  may  be  introduced  by 
allowing  the  length  to  vary  by  a  small  random  factor  e^,  which  is  referred 
to  as  disorder.  The  length  of  a  disordered  bar  is  Li  =  Zrox(l  +  £:)•  The  ratio 
of  the  connection  position  to  the  length,  aifLi^  is  held  constant.  Bar  1  is 
excited  by  a  distributed  force  Fi{xi,t). 

3.1  Nominal  transmitted  power 

The  power  transmitted  from  Bar  1  to  Bar  2  for  the  nominal  system  (no 
disorder)  is  briefly  presented  here.  A  more  detailed  derivation  is  shown  in 
the  Appendix  (see  also  Refs.  [7]  and  [8]).  The  equations  of  motion  are 

Wi(a:i,t)  -f  ^Ci(wi(a;i,f))  =  Fi(a:i,t)-|- 

^[W2(a2,  t)-wi(ai,  -  ci) 

(jBo2A2-§^  +  m2|^^W2(x2,t)  +  ^C'2(w2(a;2,t))  = 

k[wi{ai,t)-W2{a2,t)]6{x2  -  ^2) 


1182 


a) 

Bar  2 


Fig.  1:  Two-bar  system 

where  ^  is  a  Dirac  delta  function,  and  (for  Bar  i)  Eoi  is  Young’s  modulus, 
Ai  is  the  cross-sectional  area,  rm  is  the  mass  per  unit  length,  ’Ci  is  the 
viscous  damping  operator,  and  Wt(a:i,t)  is  the  deflection.  The  deflections 
of  the  two  bars  can  be  expressed  by  a  summation  of  modes: 

Wi{xi,t)  =  W2(x2,t)  =  E  1^2r(i)^2r(^2)  (5) 

t=0  r=0 

where  and  W2^{t)  are  modal  amplitudes,  and  and  ^2r(^2) 

are  mode  shape  functions  of  the  decoupled  bars.  These  mode  shape  func¬ 
tions  are  normalized  so  that  each  modal  mass  is  equal  to  the  total  mass  of 
the  bar,  Mi.  Applying  modal  analysis  and  taking  a  Fourier  transform,  the 
following  equations  are  obtained: 

Mi(f)ijWij  =  4-  fc^ij(ai)[E  kF2r^2r(<^2)  -  E 

_  (6) 

M2<f>2,W2,  =  ^^^2,(<22)[E  VFi,^i,(ai)  -  E  W2.^2M2)] 

i=0  r=0 

[ul.  -  -f  •  2Cia;i  .a;)(2  -  sgn(j)) 
y/^  •  2C2<^2.^)(2  -  sgn(s)) 

1  for  ^  >  0 
0  for  2  =  0 

where  an  over-bar  (")  denotes  a  Fourier  transform,  and  are  resonant 
frequencies,  (i  and  (2  are  damping  ratios,  and  is  a  modal  driving  force. 
Mode  0  is  a  rigid  body  mode,  which  is  why  the  sgn(i)  term  is  present. 
Note  that  the  damping  ratio  of  each  bar  is  assumed  to  be  the  same  for  all 
modes. 

Next,  the  modal  driving  forces  are  assumed  to  be  incoherent,  and  each 
spectral  density  function  is  assumed  to  be  constant  (white  noise)  over  a 
finite  range  of  frequency  [2].  After  some  algebra,  the  power  transmitted 


<^2^  = 

sgn(0  = 


1183 


(7) 


from  Bar  1  to  Bar  2,  1112(0;),  is  found  as: 

2o;^PC2«^PlPl  ^  ^2r(^2)0^2r 

M1W2IAI2  ^  |(^2.|2 

-^1  -^2  <?^2r 

where  5pjpi  is  the  same  uniform  spectral  density  function  for  each  modal 
driving  force  on  Bar  1. 


1112(0;)  = 
A  = 


3.2  Monte  Carlo  Energy  Method  (MCEM) 

The  disordered  case  is  now  considered,  where  each  bar  has  a  random  length. 
The  ensemble-averaged  transmitted  power  for  a  population  of  disordered 
two-bar  systems  is  found  by  tahing  the  expected  value  of  Eq.  (7): 


^[ni2] 


(8) 


where  Ni  is  the  number  of  modes  taken  for  Bar  i  (this  is  an  arbitrary  set  of 
modes  that  have  been  aliased  to  the  numbers  1,2,-  •  ’,Ni).  Since  a  truncated 
set  of  modes  is  used,  Eq.  (8)  is  an  approximation.  The  random  variables 
in  Eq.  (8)  are  the  resonant  frequencies  of  the  bars  (which  are  present  in 
the  terms  ^i;,  ^2r?  s,nd  A). 

Equation  (8)  may  be  solved  numerically  using  a  Monte  Carlo  method: 
the  random  variables  are  assigned  with  a  pseudo-random  number  generator 
for  each  realization  of  a  disordered  system,  and  the  transmitted  power 
is  averaged  for  many  realizations.  This  is  called  a  Monte  Carlo  Energy 
Method  (MCEM)  here.  It  may  be  used  as  a  benchmark  for  comparing  the 
accuracy  of  other  approximate  methods. 

Note  that  the  resonant  frequencies  of  a  bar  may  be  found  directly  from 
the  disordered  length.  Therefore,  for  the  MCEM  results  in  this  study,  the 
actual  number  of  random  variables  in  Eq.  (8)  is  taken  to  be  one  for  each 
decoupled  bar.  That  is,  the  two  random  lengths  are  assigned  for  each  real¬ 
ization,  and  then  the  natural  frequencies  are  found  for  each  bar  in  order  to 
calculate  the  transmitted  power.  If  such  a  relation  were  not  known,  each 
resonant  frequency  could  be  treated  as  an  independent  random  variable. 


3.3  SEA-equivalent  Transmitted  Power 

An  SEA  approximation  of  the  transmitted  power  may  be  obtained  by  ap¬ 
plying  several  typical  SEA  assumptions  to  Eq.  (8).  (Since  Eq.  (1)  is  not 
used  directly,  this  might  be  called  an  SEA-equivalent  transmitted  power.) 
These  assumptions  were  summarized  in  Ref.  [8]:  the  coupling  between 


1184 


substructures  is  weak,  the  modal  responses  are  uncorrelated,  the  expected 
value  of  the  square  of  mode  shape  functions  at  connection  positions  is  unity, 
and  the  pdfs  of  the  resonant  frequencies  are  uniform  and  identical. 

The  assumption  of  weak  coupling  means  that  the  value  of  jAj  in  Eq. 
(8)  is  approximately  one.  Applying  the  second  and  third  assumptions  then 
yields 


E[ai2] 


M?M2 


(9) 


Since  the  pdf  of  each  resonant  frequency  is  assumed  to  be  uniform,  the 
expected  values  in  Eq.  (9)  are 


E[ 


E[ 


l/4u)" 

1 

.2cos| 

-f  2a;a;i.  •  cosf  + 
^  ul.  —  2u}ujii  •  cos|  • 

•  sin  I 


sin% 


ian~ 


U) 


2  _ 


(10) 


l/4a;"C2Vr^,  -  1) 

- 1 - tan  — — - - — 


W2r=‘*'2ru 


—  ^2r 


4w^C2\/i  -  Cl 


(11) 


W2r=‘*'2rj 


where  a  =  cos~^(l  -  2Cj),  subject  to  the  restrictions  (1  -  2(1)^  <  1  and 
(1  _  2(2)2  ^  I  Finally,  since  the  pdfs  of  the  resonant  frequencies  are  taken 
to  be  identical,  the  frequency  limits  do  not  depend  on  the  individual  modes 
(a;i.  =  uj2r,  -  a^nd  =  u;2,„  =  oju)-  Therefore,  each  sum  in  Eq.  (9) 
simplifies  to  the  product  of  the  expected  value  and  the  number  of  modes 
in  the  frequency  range  of  interest: 


E[n^2] 


2C2A:ViViiV25,^p^  r.r  1  1  Pr  ^2.  ^ 

WM2  Vi.r 


(12) 


Equation  (12)  is  the  SEA  approximation  used  in  this  study. 


3.4  Example 

The  three  formulations  of  the  transmitted  power  presented  thus  far  the 
nominal  transmitted  power  in  Eq.  (7),  the  MCEM  transmitted  power  in 
Eq.  (8),  and  the  SEA-equivalent  transmitted  power  in  Eq.  (12)  —  are 
now  compared  for  a  two-bar  system  with  the  parameters  shown  in  Table  1. 
For  the  MCEM  results,  the  disorder  (ei  and  62)  was  taken  to  be  uniformly 
distributed  with  mean  zero  and  standard  deviation  ai  =  0-2  =  10%. 

As  a  measure  of  the  coupling  strength,  the  coupling  ratio,  Ri,  is  defined 
cLS  the  ratio  of  coupling  stiffness  to  the  equivalent  stiffness  of  a  bar  at  the 


1185 


Table  1:  Material  properties  and  dimensions  of  two  bars 


MIIM2 

21.53/21,53 

[Kg] 

R01IL02 

10.58/8.817 

[m] 

E01IR02 

200  XIOV2OO  xlO^ 

[N/m'1 

ai/a2 

2.116/7.053 

N 

Pllp2 

7,800/7,800 

[Kg/m'*l 

^pipi 

1 

[N^l 

C1/C2 

0.005/0.005 

k 

4.868  xlO^ 

[N/m; 

Fig.  2:  Comparison  of  the  nominal  transmitted  power,  the  MCEM  results 
(20,000  realizations  with  (j^  =  <72-  10%),  and  the  SEA  approximation. 


fundamental  resonant  frequency,  Ri  =  weak  cou¬ 

pling  is  considered  here  such  that  Ri  =  0.01. 

The  nominal  transmitted  power,  the  MCEM  results,  and  the  SEA  ap¬ 
proximation  are  shown  in  Fig.2.  The  transmitted  power  calculated  for 
the  nominal  system  exhibits  distinct  resonances.  This  is  due  to  the  low 
modal  overlap  of  the  bars  in  this  frequency  range.  The  MCEM  results 
show  distinct  peaks  for  uj  <  15,000  rad/s,  but  they  become  smooth  as 
the  frequency  increases.  The  SEA  approximation  does  not  capture  indi¬ 
vidual  resonances.  However,  at  the  higher  frequencies  where  the  disorder 
effects  are  stronger,  the  SEA  approximation  agrees  well  with  the  MCEM 
results.  The  frequency  range  between  where  the  MCEM  results  are  close 
to  the  nominal  results  and  where  they  are  close  to  the  SEA  results  (ap¬ 
proximately  2,500  ~  15,000  rad/s  for  this  case)  is  considered  to  be  the 
mid-frequency  range  here.  This  range  will  vary  depending  on  the  system 


1186 


parameters  and  the  disorder  strength.  In  the  next  section,  an  efficient  ap¬ 
proximation  of  the  transmitted  power  is  presented  which  compares  well 
with  MCEM  in  the  mid-frequency  range. 


4  Parameter-based  Statistical  Energy 
Method  (PSEM) 

The  SEA  approximation  presented  in  the  previous  section  does  not  capture 
the  resonances  in  the  transmitted  power  because  of  two  assumptions:  the 
resonant  frequencies  all  have  the  same  uniform  pdf,  and  the  values  of  the 
square  of  mode  shape  functions  at  the  connection  positions  are  taken  to  be 
the  ensemble- averaged  value.  Keane  proposed  an  alternate  pdf  of  resonant 
frequencies  in  order  to  apply  SEA  to  the  case  of  two  coupled  nearly  periodic 
structures  [9].  This  pdf  is  shown  in  Fig.3(a).  It  accounts  for  the  fact  that 


Fig.  3:  (a)  The  pdf  of  the  natural  frequencies  and  the  resultant  transmitted 
power  from  Ref.  [9].  (b)  The  pdfs  of  three  natural  frequencies,  and  a 
schematic  representation  of  the  piecewise  evaluation  of  transmitted  power 

for  PSEM.  The  individual  modal  contributions  are  extrapolated  ( - )  and 

summed  to  calculate  the  total  transmitted  power  ( — ). 

the  natural  frequencies  of  a  nearly  periodic  structure  tend  to  be  grouped  in 
several  distinct  frequency  bands.  Thus  the  pdf  has  a  large  constant  value 
for  those  frequency  bands,  and  a  small  constant  value  elsewhere.  The  SEA 
approximation  of  transmitted  power  is  then  modified  by  simply  adding  a 
positive  value  or  negative  value  on  a  logarithmic  scale,  as  demonstrated  in 
Fig,  3(a).  This  solution  thus  captures  some  of  the  resonant  behavior  of  the 
transmitted  power. 


1187 


Here,  a  more  general  approach  is  taken  for  approximating  the  power 
transmitted  between  two  substructures  in  a  frequency  range  in  which  they 
have  low  or  intermediate  modal  overlap.  Each  resonant  frequency  is  as¬ 
signed  a  uniform  pdf.  However,  the  frequency  range  of  each  pdf  is  diiferent; 
it  corresponds  to  the  range  in  which  that  resonant  frequency  is  most  likely 
to  be  found.  (The  concept  of  using  “confidence  bands”  as  one-dimensional 
pdfs  was  suggested  but  not  pursued  in  Ref.  [8].)  An  example  is  shown  in 
Fig.  3(b)  for  three  resonant  frequencies.  Furthermore,  it  is  assumed  that 
the  values  of  the  square  of  the  mode  shape  functions  at  the  connection 
positions  are  known.  Thus,  applying  only  the  first  two  SEA  assumptions 
along  with  those  noted  above,  Eq.  (8)  becomes: 


N2 

i:kme 


This  is  called  a  Parameter-ba.sed  Statistical  Energy  Method  (PSEM)  be¬ 
cause  it  employs  information  for  individual  modal  parameters. 

Since  each  modal  pdf  is  uniform,  Eqs.  (10)  and  (11)  still  hold  for  the 


expected  values  in  Eq.  (13).  However,  unlike  the  SEA  approximation,  each 
expected  value  is  different,  because  the  corresponding  frequency  bounds  are 
unique.  Furthermore,  note  that  the  pdfs  do  not  cover  the  entire  frequency 
range  of  interest.  The  results  for  each  mode  are  therefore  extrapolated 
outside  the  frequency  range  of  that  modal  pdf  before  the  individual  modal 
contributions  are  summed.  This  is  shown  schematically  in  Fig.  3(b).  PSEM 


is  therefore  a  piecewise  evaluation  of  the  expected  value  of  transmitted 


power. 

The  PSEM  approximation  is  now  applied  to  the  two-bar  system  of 
Table  1,  with  the  standard  deviation  of  disorder  cti  =  0-2  =  10%.  The  pdfs 
of  the  resonant  frequencies  of  Bar  1  and  Bar  2  aze  shown  in  Fig.  4(a)  and 
(b),  respectively.  For  this  system,  the  bounds  for  each  resonant  frequency 
may  be  found  directly  from  the  variation  of  the  uncertain  parameter.  It 
can  be  seen  that  the  spread  of  each  natural  frequency  pdf  due  to  disorder 
increases  with  increasing  frequency. 

The  MCEM,  PSEM,  and  SEA  approximations  for  the  transmitted  power 
are  shown  in  Fig.  4(c).  There  were  20,000  realizations  taken  for  the  MCEM 
results  at  each  sampled  frequency.  This  took  about  10  hours  of  computa¬ 
tion  time.  In  contrast,  the  PSEM  results  only  required  3  seconds  of  compu¬ 
tation  time,  and  the  SEA  results  only  required  about  1  second.  Note  that 
the  PSEM  results  show  excellent  agreement  with  the  much  more  expensive 
MCEM  results.  The  difference  at  very  low  frequencies  comes  from  the  fact 
that  for  the  PSEM  approximation,  the  value  of  the  term  |A|  was  assumed 
to  be  one  due  to  weak  coupling.  This  assumption  breaks  down  as  the 
frequency  approaches  zero.  However,  the  match  between  the  MCEM  and 


1188 


Fig.  4:  (a)  Natural  frequency  pdfs  for  Bar  1.  (b)  Natural  frequency  pdfs 
for  Bar  2.  (c)  Transmitted  powers  obtained  by  MCEM,  PSEM,  and  SEA 
for  c7i  =  <J2  =  10%. 

PSEM  results  in  the  mid-frequency  range  is  excellent.  Again,  it  is  noted 
that  the  SEA  results  converge  to  those  of  MCEM  (and  PSEM)  as  the  fre¬ 
quency  increases.  Now  it  can  be  seen  that  the  assumption  of  identical  pdfs 
for  all  modes  becomes  better  with  increasing  frequency. 

Next,  the  example  system  is  considered  with  smaller  disorder,  Ci  = 
(72  =  1%.  Fig.  5  shows  the  results  for  this  case  for  what  might  be  called 
the  mid-frequency  range.  Note  that  even  though  this  is  a  higher  frequency 
range  than  that  considered  for  the  previous  ca.se,  the  pdfs  of  the  resonant 
frequencies  shown  in  Fig.  5(a)  and  (b)  are  not  as  strongly  overlapped. 
Thus,  several  peaks  are  seen  in  the  transmitted  power  in  Fig.  5(c).  Again, 
the  PSEM  approximation  agrees  well  with  the  MCEM  results,  although 
there  is  more  discrepancy  for  this  case.  The  SEA  approximation  follows 
the  global  trend,  but  does  not  capture  the  resonances  or  anti-resonances. 
The  SEA  results  drop  off  at  the  edges  because  only  modes  within  this 
frequency  range  are  considered  to  contribute  to  the  transmitted  power. 

In  addition  to  PSEM,  another  piecewise  evaluation  of  the  transmitted 
power  is  considered  here.  For  this  approximation,  wherever  the  individual 
mode  pdfs  overlap,  they  are  superposed  to  form  a  pdf  for  all  the  modes  in 
that  “section”  of  the  frequency  range.  This  superposition  is  demonstrated 
in  Fig.  6.  Also,  if  the  number  of  modes  in  a  section  is  above  a  certain  cutoff 
number,  Nc,  then  it  is  assumed  that  their  mode  shape  function  values  at 
the  connection  positions  are  unknown,  so  that  the  ensemble-averaged  value 
must  be  used.  This  is  called  a  multiple  mode  approximation.  The  purpose 


1189 


Fig.  5:  (a)  Natural  frequency  pdfs  for  Bar  1.  (b)  Natural  frequency  pdfs 
for  Bar  2.  (c)  Transmitted  power  obtained  by  MCEM,  PSEM,-and  SEA 
for  cTi  =  0-2  =  1%. 


Fig.  6:  Resonant  frequency  pdfs  for  PSEM  and  for  the  multiple  mode 
approximation. 

of  formulating  this  approach  is  to  investigate  what  happens  as  information 
about  the  individual  modes  is  lost. 

The  multiple  mode  approximation  is  applied  to  the  example  system 
with  cTi  =  <72  =  10%  in  Fig. 7.  For  A/c  =  2,  this  approximation  has  good 
agreement  with  MCEM.  The  match  is  especially  good  for  u)  <  10,000. 
Above  this  frequency,  the  number  of  overlapped  resonant  frequencies  in 
each  pdf  section  is  greater  than  Nc,  and  the  loss  of  mode  shape  information 
affects  the  results  slightly.  For  Nc  =  0,  the  values  of  the  mode  shape 
functions  are  taken  to  be  one  for  the  entire  frequency  range,  just  as  in  the 
SEA  approximation.  As  can  be  seen  in  Fig.  7,  the  piecewise  construction 
of  the  pdf  roughly  captures  the  frequency  ranges  of  the  resonances  and 
anti-resonances.  However,  the  mode  shape  effect  is  more  pronounced  in 


1190 


-B.S 


-  MCEM 

-e.s  —  Multiple  mode  approximation  rvt=  2 

.  Multiple  mode  approximation  Nfc=  O 


Fig.  7:  Transmitted  power  obtained  from  MCEM  and  the  multiple  mode 
approximation  for  Ui  =  <72  =  10% 

the  mid-frequency  range.  The  peak  values  are  now  similar  to  the  SEA 
approximation. 


5  Conclusions 

In  this  study,  the  power  transmitted  between  two  multi-mode  substruc¬ 
tures  coupled  by  a  spring  was  considered.  A  Monte  Carlo  Energy  Method 
(MCEM)  was  used  to  calculate  the  ensemble  average  of  the  transmitted 
power  for  the  system  with  parameter  uncertainties.  A  classical  Statistical 
Energy  Analysis  (SEA)  approximation  matched  the  Monte  Carlo  results 
in  the  high-frequency  range,  but  did  not  capture  the  resonant  behavior  of 
the  transmitted  power  in  the  mid-frequency  range  where  the  substructures 
have  weak  modal  overlap. 

A  Parameter-based  Statistical  Energy  Method  (PSEM)  was  presented 
which  uses  a  distinct  pdf  for  each  natural  frequency  as  well  as  some  indi¬ 
vidual  mode  shape  information.  A  piecewise  evaluation  of  the  transmitted 
power  was  performed,  and  then  the  modal  contributions  were  extrapolated 
and  superposed.  The  PSEM  approximation  compared  very  well  with  the 
much  more  expensive  Monte  Carlo  results,  including  in  the  mid-frequency 


1191 


range. 


References 


1  R.  H.  Lyon.  Statistical  Energy  Analysis  of  Dynamical  Systems:  Theory 
and  Applications.  M.I.T.  Press,  1st  edition,  1975. 

2  R.  H.  Lyon.  Theory  and  Application  of  Statistical  Energy  Analysis. 
Butterworth-Heinemann,  2nd  edition,  1995. 

3  C.  H.  Hodges  and  J.  Woodhouse.  Theories  of  noise  and  vibration  trans¬ 
mission  in  complex  structures.  Rep. Prog. Physics,  49:107-170,  1986. 

4  J.  Woodhouse.  An  approach  to  the  theoretical  background- of  statis¬ 
tical  energy  analysis  applied  to  structural  vibration.  Journal  of  the 
Acoustical  Society  of  America,  69(6):1695-1709,  1981. 

5  M.  P.  Norton.  Fundamentals  of  Noise  and  Vibration  Analysis  for  En¬ 
gineers.  Cambridge  University  Press,  1st  edition,  1989. 

6  N.  Lalor.  Statistical  energy  analysis  and  its  use  as  an  nvh  analysis  tool. 
Sound  and  Vibration,  30(l):16-20,  1996. 

7  Huw  G.  Davies.  Power  flow  between  two  coupled  beams.  Journal  of 
the  Acoustical  Society  of  America,  51(1):393-401,  1972. 

8  A.  J.  Keane  and  W.  G.  Price.  Statistical  energy  analysis  of  strongly 
coupled  systems.  Journal  of  Sound  and  Vibration,  117(2):363-386, 
1987. 

9  A.  J.  Keane.  Statistical  Energy  Analysis  of  Engineering  Structures 
(Ph.D  Dissertation).  Brunei  University  (England),  1988. 


Appendix 


In  this  appendix,  the  nominal  transmitted  power  in  Eq.  (7)  for  the  mono- 
coupled  two-bar  system  is  derived.  The  procedure  follows  that  of  Refs  [7, 
8]. 

Plugging  Eq.  (5)  into  Eq.  (4), 


F^ixut)  +  fcf  E  -  Oi)  (A.1) 


t=0 


1192 


k  £  Wi,(<)$i,(a,)  -  £  lV2,(«)«2,(a2)  5(x2  -  aj) 


Multiplying  Eq.  (A.l)  by  and  integrating  with  respect  to  xi  for 

[0,  Li]  yields 


MiK  +  = 

A,  +  f f;  W2,(4)W2Xo2)  -  E 


Li 

2CijUijMi  =  y  C'i(«'i,.(a:i))Wij(xi)cia:i  ,  fi,  J  Fi{xi,t)^i.{xi)dx], 

0  0 

and  wij  is  the  ;th  resonant  frequency  of  decoupled  Bar  1.  The  damping 
ratio  in  Eq.  (A. 3)  is  now  assumed  to  be  the  same  (Ci)  for  all  modes,  since 
the  differences  in  the  ratio  are  usually  small  and  this  simplifies  the  equation. 
Taking  the  Fourier  transform  of  Eq.  (A.3)  with  zero  initial  conditions  leads 
to  the  following 

=  7i, +  S:$i,(ai)fETr2, $2,(02) 

t  r=0  i=0  -* 

</>!;•  =  K,  -  •  2(iu;i^iv){2  -  sgn(i)) 

J  fori  =  0 

where  (~)  denotes  a  Fourier  transform.  Similarly,  applying  the  previous 
procedure  to  Eq.  (A.2), 

r  OO  CO  -1 

M2(^2.1V2.  =  ^^$2.(02)  EW"li®l.(<»0-EW"2,$2,(o2)  (A.5) 

t  i=0  r=0 

^2s  =  {^2,  -  •  2C2^^2,w)(2  -  sgn(s)). 

Solving  for  VF2,  from  Eq.  (A.4)  and  (A.5), 

=  Trr-Wx-l^^-  -  (a.6) 


where 


_  W2.(a2).^/ii$i,(ai) 


1193 


Calculating  the  second  term  in  brackets  in  Eq.  (A. 6), 
fc4'2.(<22)EW^2,«'2.(a2)  E 


where 


Plugging  Eq.  (A. 7)  into  Eq.  (A. 6), 

M2<j>2s  1  +  CKl  +  Q;2  1=0 

\  I '  I.  -I—  ✓ 

Coupling  force  Ps 

Using  the  definition  of  transmitted  power  in  Ref.  [7],  1112(0))  is 


ni2  =  Re[-v^o)f;E[P.F;,(o))] 

L  „_ri 


"I"  0^1  +  P  i~0  j=:0  J 

A 

i<?^2.p  hh  Ml, 

where  Re[]  denotes  the  real  part  of  the  argument,  *  is  a  complex  conjugate, 
and 

Finally,  it  is  assumed  that  the  modal  driving  forces,  f-^.,  are  uncorrelated. 
Also,  the  spectral  density  function  of  each  modal  driving  force  is  assumed 
to  be  constant  for  the  finite  frequency  range  of  interest: 


C  (  \  f  ^PlVl  ^ 

V  for  i^j. 


Therefore,  Eq.  (A. 9)  becomes 


2^2fc^O)^.5^pipi  ^  ^  ^2r(^2)o)2r 

to  to 


(A.IO) 


(A.ll) 


1194 


PASSIVE  AND  ACTIVE  CONTROL  III 


Research  on  Control  Law  of  Active  Siispension  of  a 
Seven  Degree  of  Freedom  Vehicle  Model 

Dr&Prof.  YuchengLei  Lifen  Chen 
Automobile  Engineering  Dept,  Tong  Ji  University  ,Shang 
Hai,P.R.of  China 

Abstract 

In  the  paper  ,  control  law  of  active  suspension  is  presented  , 
which  involves  7-DOF  vehicle  model  for  improving  control 
accuracy  .The  control  law  involve  vehicle  running  velocity  , 
road  power  spectrum  ,  suspension  stiffness  and  damping  .The 
control  law  can  be  applied  to  multi-DOF  control  of  active 
suspension  of  vehicle  . 

Keywords:  Active  suspension  ,  control  law  ,  Game  theory  , 
Modeling  ,  7  —  DOF  Vehicle  Model . 

1 .  Introduction 

An  individual  control  system  for  each  wheel  by  applying  the 
optimum  regulation  method  for  the  two  degrees  of  freedom  is 
showed  in  [3]  .  [4]  and  [5]  also  introduce  two-DOF  feedback 
control  method  of  active  suspension  .It  is  difficult  for  two- 
DOF  control  method  to  coordinate  multi-DOF  kinematic 
distances  of  entire  car  .  Muti-DOF  active  control  can  improve 
coordination  control  accuracy  of  entire  car ,  but  high  speed  of 
CPU  is  asked  for  control  and  calculation  while  control  law  of 
multi-DOF  is  got  by  real-time  calculation  .  And,  ride 
performance  and  handling  performance  is  inconsistent  .  For 
resolving  the  problem  ,the  paper  holds  a  new  calculation 
method  for  optimizing  the  law  that  can  be  programmed  for 
real-time  control  by  table-lookup  and  not  by  real-time 
calculation  .So  the  method  and  law  can  not  only  improve 
coordination  control  accuracy  ,but  also  develop  control 
speed . 

2 .  Mathematical  Model 

Vehicle  is  simplified  to  turn  into  7  DOF  model  .  7-DOF 
vibration  motion  equation  can  be  written  as  follow 

[Af  ]z+  [c]z+  [kY  =  \C,  ](2+  ]e  (1) 


1195 


Where  [M]  is  mass  matrix  ,  [C]  is  suspension  damping 

matrix  ,  [A’]  is  suspension  stiffness  matrix  ,  [C,]  is  tyre 

•  • 

damping  matrix  ,  \K,  ]  is  tyre  stiffness  matrix  ,  Z  is 
acceleration  matrix  ,Z  is  velocity  matrix  ,  Z  is  7-DOF 


displacement  matrix ,  Q  is  road  surface  input  velocity  matrix , 
Q  is  road  surface  input  displacement  matrix . 

z=p„z....,zy  (2) 


••  .  •  -j-J 

Where  Zi  is  vehicle  vertical  acceleration,  Zi  is  roll 

••  •• 

acceleration  ,  Z3  is  pitch  acceleration  ,  Z,-  (/  =  4,*«*,7)  is 

four  tyres  vertical  acceleration  . 

3  General  Optimization  Method  of  Control  Law 
Objective  function  of  optimization  of  control  law  can 
generally  got  by  calculating  weighted  sum  of  7-DOF  mean 
square  root  of  acceleration  ,  dynamic  deflection  and  dynamic 
load  .it  can  be  written  as  follows 

/=!  >1  A=1 

Where  a-^  (z  =  !,•  •  (7  =  !,•  •  •A),  r  *  =  h*  •  •A)  is 

weighted  ratio  .  Where  a..  (i=l,  *  *  '  ,7)  is  7-DOF  mean 

Zi 

square  root  of  acceleration  ,  o-jy_  0=1?  ‘  '  is  7-DOF  mean 
square  root  of  dynamic  deflection  ,  (k=l,  *  *  ^  A)  is 

7-DOF  mean  square  root  of  relative  dynamic  load  .cr.. , 

Zi 

can  be  calculated  by  resolving  (1)  using  numerical 

method . 

4  Result  of  General  Optimization 
Method  of  Control  Law 

Optimization  result  of  control  law  of  a  truck  is  got  using 
above  method  as  figure  1  and  2  ,  its  main  parameters  as 
follows . 

Wheel  distance  is  1.4  meter  ,  axle  distance  is  2.297  meter  , 
mass  is  1121.3  kg  ,  front  tyre  and  axle’ s  mass  is  22.8  kg  , 
rear  tyre  and  axle’ s  mass  is  35.0  kg  ,  X  axis’  rotational 


1196 


inertia  is  307.4  kg-ni^  ,  Y  axis’  rotational  inertia  is  1276.5 
kg-rri^  . 

In  fig.  1  and  2  ,  Cl  of  RMSMIN  and  C3  of  RMSMIN  are 
respectively  front  and  rear  suspension  damping  of  getting 
minimization  of  above  objective  function  ,  it  is  changing 
while  road  surface  rough  coefficient  and  automobile 

velocity  V  is  changing  .  Cl  and  C3  also  rise  when  velocity  V 
rises  .  This  is  called  control  law  of  general  optimization 
method  of  active  suspension  in  the  paper  .  The  result  in  fig.  1 
and  fig.  2  has  been  verified  by  road  test . 


Fig.  1  front  suspension  Fig.  2  rear  suspension 

optimization  damping  optimization  damping 

Simulation  result  can  also  verify  that  ride  performance’s 
increasing  (suspension  stiffness  reducing)  will  make  handling 
performace  reduce  .  So  selecting  perfect 

=  is  very  difficult 

and  inconsistent  .The  paper  advances  next  game  method  to 
try  to  resolve  the  inconsistent  problem  . 

5  Game  Optimization  Method  of  Control  Law 
Because  to  select  weighted  ratio  of  general  optimization 
method  is  difficult  ,  the  paper  advances  a  new  method  of 
optimization  of  control  law  —  Game  Balance  Optimization 
Method  4t  is  discussed  as  follows  . 

Game  theory  method  of  two  countermeasure  aspect  can  be 
expressed  as  follows : 


1197 


—  1  j  ^  0  ,z  •—  1,  2, 

tr  (4) 

^  hj  —  I  ,  hj  ^0  ,  y  =  1,  2,-  •  *,  WZ2 

where  . r^,  is  probability  of  selecting  R^,R2> . of 

countermeasure  R  (  where  R  is  acceleration  mean  square 

root  )  ,  and  h,,h^, . X,  is  probability  of  selecting 

. ,  of  countermeasure  H  (  where  H  is 

mean  square  root  of  deflection  or  handling  and 

satiability  )  . 

It  is  called  hybrid  game  method  while  these  probability  is 
leaded  into  the  method .  Countermeasure  R  selects  in  order 
to  get  maximization  of  minimization  paying  expected  value 
of  column  vector  of  paying  matrix  ,  and  countermeasure  H 
selects  hj  in  order  to  get  minimization  of  maximization 
paying  expected  value  of  row  vector  of  paying  matrix  . 


If  rank  of  paying  matrix  is  x  ,R  should  select  r.,  as 
follows. 


max\min\ 


(5) 


relative 


And  H  should  select  hj  as  follows  : 


MAXl  MINI 


j=i 


(6) 


relative  hj 

a.j  (z  =  l, . ,y  =  l, . ,^2)  in  (5)  and  (6)  is  element 

value  of  paying  matrix  ,  basing  vehicle  theory  it  can  be  got 
as  follows  : 


a.j  =  C,  !  +  (7) 

Where  C^,C^  in  (7)  is  coefficient  of  paying  matrix  (  The 
paper  orders  they  is  1  as  an  example  ,  as  Q  ,C2  ’  s  real  value 
about  very  much  condition  is  related  to  some  privacy 


1198 


problem  it  can’ t  be  introduced. )  .  (5), (6)  called  respectively 
minimization  maximization  expected  value  and  maximization 
minimization  expected  value  can  be  abbreviated  as 
MAxi^IN^^  )}  and  MIN  ^AX(^  )}  .  if  r.,andhj  is 

got  as  optimization  of  countermeasure  ,  it  can  be  wntten  as 
follows  : 

MAX^IN^  ^^Optimization  Countermeasure 

Expected  Value  ^  MIN 

A  probability  association  {r^ Xj)  can  be  content  with 

optimization  expected  value  as  follows  : 

Optimization  Countermeasure  Expected 

m^  m-i 

Value=|]  J^aij»r;»hj  (9) 

/=! 

Writer  advances  reformation  simplex  algorithm  for  resolving 
the  game  problem  as  reference  [1]  .  In  the  paper  the  water 
selects  only  an  example  to  introduce  calculation  results  as 
follows  because  the  paper  has  limited  space  . 

6  Results  of  Game  Optimization  Method  of  Control  Law 
Paying  matrix  as  fig.  3  and  fig.  4  ,  optimization  result  of 
control  law  of  a  truck  is  got  using  above  game  method  as 
figure  5  and  6  ,  calculated  truck’s  main  parameters  as 
follows . 

Wheel  distance  is  1.23  meter  ,  axle  distance  is  3.6  meter  , 
mass  is  13880.0  kg  ,  front  tyre  and  axle’ s  mass  is  280.0  kg  , 
rear  tyre  and  axle’ s  mass  is  280.0  kg  ,  X  axis’  rotational 
inertia  is  1935  kg^m^  ,  Y  axis’  rotational  inertia  is  710 
kg-m-  . 

In  fig.  3  and  4  3(IJ)  is  paying  matrix  value 

{i  =  1, . ,m,  J  =  1, . ,m,) .  In  fig.  5  and  fig.  6  ,  K1  of 

RMSMIN  and  K3  of  RMSMIN  are  respectively  front  and 
rear  suspension  stiffness  of  getting  optimization 
countermeasure  expected  value  of  above  game  method  ,  it  is 
changing  while  road  surface  rough  coefficient  Q  and 

automobile  velocity  V  is  changing  .  K1  and  K3  also  rise 


1199 


when  velocity  V  rises  .  This  is  called  control  law  of  active 
suspension  .  The  result  in  fig.  5  and  fig.  6  has  been  verified 
by  road  test . 


Fig.  3  paying  matrix  Fig.  4  paying  matrix 

7  Conclusion 

The  paper  introduces  two  method  to  get  optimization  control 


Fig.  5  front  suspension  Fig.  6  rear  suspension 

optimization  stiffness  optimization  stiffness 

law  of  active  suspension  ,  and  the  control  law  is  verified  to 
ability  to  be  applied  to  real  control  of  active  suspension  .  This 
will  develop  control  accuracy  and  speed  of  active 
suspension  .Off  course  ,  it  need  being  researched  further  . 

8  reference 

1  Lei  Yucheng  ,  Theory  and  Engineering  Realization  of 
Semi  —  Active  Control  of  Vehicle  Vibration  ,Dr.  paper  , 
Harbin  Institute  of  Technology , China  ,1995  6  . 

2  Thompson  A.G.  ,  A  Suspension  Proc.  Int  of  Mechanical 


1200 


Engr.  Vol  185  No. 36, 970  —  990,553  563  . 

3  Lei ,  S. ,  Fasuda  ,  E.  and  Hayashi ,  Y.  :  “An  Experimental 
Study  of  Optimal  Vibration  Adjustment  Using  Adaptive 
Control  Methods  ”,  Proc  .  IMechE  Int  .  Conf  .  Advanced 
Suspensions  ,  London  ,  England  ,  (1988)  ,  C433/88  ,  119- 
124. 

4  Kamopp  D.  , Active  Damping  in  Road  Vehicle  Systems  , 
VSD,  12(1983),  291-316. 

5  Kamopp  D  .  C  .,  Grosby  M.  J.  &  Harword  R.  ,Vibration 
Control  Using  Semi-Active  Force  ,  Generator  ,  Trans  . 
ASME,  J  .  Eng  .  for  Ind  .  Vol .  96  Ser  .  B  ,  No  .2  ,  (1974) , 
619-626 . 


1201 


1202 


Designing  Heavy  Truck  Suspensions  for  Reduced  Road  Damage 

Mehdi  Ahmadian 
Edward  C.  Mosch  Jr. 


Department  of  Mechanical  Engineering 
Virginia  Polytechnic  Institute  and  State  University 
Blacksburg,  Virginia  24061-0238;  USA 
(540)  231-4920/-9100(fax) 
ahmadian@vt.edu 


ABSTRACT 

The  role  of  semiactive  dampers  in  reducing  tire  dynamic  loading  is  examined. 
An  alternative  to  the  well-known  skyhook  control  policy,  called 
“groundhook,”  is  introduced.  Using  the  dynamic  model  of  a  single  suspension, 
it  is  shown  that  groundhook  semiactive  dampers  can  reduce  tire  dynamic 
loading,  and  potentially  lessen  road  damage,  for  heavy  trucks. 


INTRODUCTION 

The  main  intent  of  this  work  is  to  determine,  analytically,  the  role  of 
semiactive  suspension  systems  in  reducing  tire  dynamic  loading,  and  road  and 
bridge  damage.  Although  primary  suspension  systems  with  semiactive 
dampers  have  been  implemented  in  some  vehicles  for  improving  ride  and 
handling,  their  impact  on  other  aspects  of  the  vehicle  remain  relatively 
unknown.  Specifically,  it  is  not  yet  known  if  implementing  semiactive 
dampers  in  heavy  truck  suspension  systems  can  reduce  the  tire  dynamic 
forces  that  are  transferred  by  the  vehicle  to  the  road.  Reducing  dynamic  forces 
will  result  in  reducing  pavement  loading,  and  possibly  road  and  bridge  damage. 

The  idea  of  semiactive  dampers  has  been  in  existence  for  more  than 
two  decades.  Introduced  by  Karnopp  and  Crosby  in  the  early  70’ s  [1-2], 
semiactive  dampers  have  most  often  been  studied  and  used  for  vehicle  primary 
suspension  systems.  A  semiactive  damper  draws  small  amounts  of  energy  to 
operate  a  valve  to  adjust  the  damping  level  and  reduce  the  amount  of  energy 
that  is  transmitted  from  the  source  of  vibration  energy  (e.g.,  the  axle)  to  the 
suspended  body  (e.g.,  the  vehicle  structure).  Therefore,  the  force  generated  by 
a  semiactive  damper  is  directly  proportional  to  the  relative  velocity  across  the 
damper  (just  like  a  passive  damper).  Another  class  of  dampers  that  is  usually 
considered  for  vibration  control  is  fully  active  dampers.  Active  dampers  draw 

1203 


relatively  substantial  amounts  of  energy  to  produce  forces  that  are  not 
necessarily  in  direct  relationship  to  the  relative  velocity  across  the  damper. 

The  virtues  of  active  and  semiactive  dampers  versus  passive  dampers 
have  been  addressed  in  many  studies  [3-10].  Using  various  analytical  and 
experimental  methods,  these  studies  have  concluded  that  in  nearly  all  cases 
semiactive  dampers  reduce  vibration  transmission  across  the  damper  and 
better  control  the  suspended  (or  sprung)  body,  in  comparison  to  passive 
dampers.  Further,  they  have  shown  that,  for  vehicle  primary  suspension 
systems,  semiactive  dampers  can  lower  the  vibration  transmission  nearly  as 
much  as  fully  active  dampers;  without  the  inherent  cost  and  complications 
associated  with  active  dampers.  This  has  led  to  the  prototype  application, 
and  production,  of  semiactive  dampers  for  primary  suspensions  of  a  wide 
variety  of  vehicles,  ranging  from  motorcycles,  to  passenger  cars,  to  bus  and 
trucks,  and  to  military  tanks,  in  favor  of  fully  active  systems. 

Although  there  is  abundant  research  on  the  utility  of  semiactive 
dampers  for  improving  vehicle  ride  and  handling,  their  potential  for  reducing 
dynamic  forces  transmitted  to  the  pavement  remains  relatively  unexplored. 
This  is  because  most  suspension  designers  and  researchers  are  mainly 
concerned  with  the  role  of  suspension  systems  from  the  vehicle  design 
perspective.  Another  perspective,  however,  is  the  effect  of  suspension 
systems  on  transmitting  dynamic  loads  to  the  pavement. 


ROAD  DAMAGE  STUDIES 

Dynamic  tire  forces,  that  are  heavily  influenced  by  the  suspension,  are 
believed  to  be  an  important  cause  of  road  damage.  Cole  and  Cebon  [11] 
studied  the  design  of  a  passive  suspension  that  causes  minimum  road  damage 
by  reducing  the  tire  force.  They  propose  that  there  is  a  stronger  correlation 
between  the  forth  power  of  the  tire  force  and  road  damage  than  the  dynamic 
load  coefficient  (DLC)  and  road  damage. 

A  simple  measure  of  road  damage,  introduced  by  Cebon  m  [12],  is  the 
aggregate  fourth  power  force  defined  as 

Na 

Al=Y,Pjk  k=  1,2,3  ..ns  (1) 

where  Pjk  =  force  applied  by  tire  j  to  point  k  along  the  wheel  path, 
ria  -  number  of  axles  on  vehicle,  and 

Us  =  number  of  points  along  the  road.. 


1204 


DLC  is  a  popular  measurement  frequently  used  to  characterize  dynamic 
loading  and  is  defined  as  the  root  mean  square  (RMS)  of  the  tire  force  divided 
by  the  mean  tire  force,  which  is  typically  the  static  weight  of  the  vehicle.  The 
equation  takes  the  form; 


RMS  Dynamic  Tire  Force 
Static  Tire  Force 

This  study  shows  that  minimum  road  damage,  for  a  two  degree-of-freedom 
model,  is  achieved  by  a  passive  system  with  a  stiffness  of  about  one  fifth  of 
current  air  suspensions  and  a  damping  of  about  twice  that  typically  provided. 
In  practice,  however,  reducing  the  suspension  stiffiiess  can  severely  limit  the 
static  load  carrying  capacity  of  the  suspension  and  cause  difficulties  in  vehicle 
operation.  Further,  higher  damping  can  substantially  increase  vibration 
transmission  to  the  body  and  worsen  the  ride. 

In  another  study  by  Cole  and  Cebon  [13]  a  two-dimensional  articulated 
vehicle  simulation  is  validated  with  measurements  from  a  test  vehicle.  The 
effect  of  modifications  to  a  trailer  suspension  on  dynamic  tire  forces  are 
investigated.  The  RMS  of  dynamic  loads  generated  by  the  trailer  are  predicted 
to  decrease  by  3 1  per  cent,  resulting  in  a  predicted  decrease  in  theoretical  road 
damage  of  about  13  per  cent. 

Yi  and  Hedrick  compared  the  effect  of  continuous  semiactive  and 
active  suspensions  and  their  effect  on  road  damage  using  the  vehicle  simulation 
software  VESYM  [14].  A  control  strategy  based  on  the  tire  forces  in  a  heavy 
truck  model  is  used  to  show  that  active  and  semiactive  control  can  potentially 
reduce  pavement  loading.  They,  however,  mention  that  measuring  the  tire 
forces  poses  serious  limitation  in  practice. 

The  primary  purpose  of  this  paper  is  to  extend  past  studies  on 
semiactive  suspension  systems  for  reducing  road  damage.  An  alternative 
semiactive  control  policy,  called  "groundhook,”  is  developed  such  that  it  can 
be  easily  applied  in  practice,  using  existing  hardware  for  semiactive 
suspensions.  A  simulation  model  representing  a  single  primary  suspension  is 
used  to  illustrate  the  system  effectiveness.  The  simulation  results  show  that 
groundhook  control  can  reduce  the  dynamic  load  coefficient  and  fourth  power 
of  tire  force  substantially,  without  any  substantial  increase  in  body 
acceleration. 


1205 


MATHEMATICAL  FORMULATION 


We  consider  a  model  representing  the  dynamics  of  a  single  primary  suspension 
in  a  heavy  truck,  as  shown  in  Figure  1. 


Truck  Mass 


Suspension 

Stiffness 

Suspension 
and  Tire  Mass 


Tire 

Stiffness 


Suspension 

Damping 


Road  Input 


Figure  1.  Primary  Suspension  Model 

This  model  has  been  widely  used  in  the  past  for  automobile  applications,  due 
to  its  effectiveness  in  analyzing  various  issues  relating  to  suspensions. 
Although  it  does  not  include  the  interaction  between  the  axles  and  the  truck 
frame  dynamics,  the  model  still  can  serve  as  an  effective  first  step  in  studying 
fundamental  issues  relating  to  truck  suspensions.  Follow  up  modeling  and 
testing,  using  a  full  vehicle,  is  needed  to  make  a  more  accurate  assessment. 


The  dynamic  equations  for  the  model  in  Figure  1  are: 


M^x^+C(x^-X2)  + K(x^~X2)  =  0  (3a) 

MjX,  - C(ii  -X2)-K{x^  - ^2)  +  0 

The  variables  Xi  and  X2  represent  the  body  and  axle  vertical  displacement, 
respectively.  The  variable  Xjn  indicates  road  input,  that  is  assumed  to  be  a 
random  input  with  a  low-pass  (0  -  25  Hz)  filter.  The  amplitude  for  Xjn  is 
adjusted  such  that  it  creates  vehicle  and  suspension  dynamics  that  resembles 
field  measurements.  Such  a  function  has  proven  to  sufficiently  represent 
actual  road  input  to  the  vehicle  tires. 

Table  1  includes  the  model  parameters,  that  are  selected  to  represent  a 
typical  laden  truck  used  in  the  U.S.  The  suspension  is  assumed  to  have  a 
linear  stiffness  in  its  operating  range.  The  damper  characteristics  are  modeled 
as  a  non-linear  function,  as  shown  in  Figure  2. 


1206 


Table  1.  Model  Parameters 


Description 

Symbol 

Value 

Body  Mass 

Ml 

287  kg 

Axle  Mass 

M2 

34  kg 

Suspension  Stiffness 

K1 

196,142  N/m 

Tire  Stifftiess 

Kt 

1,304,694  N/m 

Suspension  Damping 

C 

See  Table  2 

The  bilinear  function  in  Figure  2  represents  the  force-velocity 
characteristics  of  an  actual  truck  damper.  The  parameters  selected  for  both 
passive  and  semiactive  dampers  are  shown  in  Table  2.  These  parameters  are 
selected  based  on  truck  dampers  commonly  used  in  the  U.S.  Although  we 
examined  the  effect  of  damper  tuning  on  dynamic  loading,  if  falls  outside  the 
scope  of  this  paper.  Instead,  we  concentrate  here  on  comparing  different 
semiactive  dampers  with  a  passive  damper,  using  the  baseline  parameters 
shown  in  Table  2.  The  semiactive  and  passive  damper  characteristics  used  for 
this  study  are  further  shown  in  Figure  3. 


1207 


Force  Velocity  Curve 


Figure  3.  Passive  and  Semiactive  Damper  Characteristics 


SKYHOOK  CONTROL  POLICY 

As  mentioned  earlier,  the  development  of  semiactive  dampers  dates  back  to 
early  70’s  when  Kamopp  and  Crosby  introduced  the  skyhook  control  policy. 
For  the  system  shown  in  Figure  1,  skyhook  control  implies 

X,(X]  -  ;C2 )  -  0  C  =  Con 
X](X^-X2)<0  C  =  Coff 

Where  x,  and  represents  the  velocities  of  Mi  (vehicle  body)  and  M2  (axle), 
respectively.  The  parameters  Con  and  Coff  represent  the  on-  and  off-state  of 
the  damper,  respectively,  as  it  is  assumed  that  the  damper  has  two  damping 
levels.  In  practice,  this  is  achieved  by  equipping  the  hydraulic  damper  with  an 
orifice  that  can  be  driven  by  a  solenoid.  Closing  the  orifice  increases  damping 
level  and  achieves  Con,  whereas  opening  it  gives  Coff . 


1208 


Table  2.  Damper  Parameters 


Passive 

Semiactive  On-State 

Semiactive  Off-State 

n 

0.25 

0.35 

0.03 

m 

0.10 

0.15 

0.03 

Vbb 

0.254  m/sec 

0.254  m/sec 

0.254  m/sec 

B 

0.20 

0.30 

0.03 

S4 

0.10 

0.15 

0.03 

Vbj 

0.254  m/sec 

0.254  m/sec 

0.254  m/sec 

The  switching  between  the  two  damper  states,  shown  in  Eq.  (4),  is 
arranged  such  that  when  the  damper  is  opposing  the  motion  of  the  sprung 
mass  (vehicle  body),  it  is  on  the  on-state.  This  will  dampen  the  vehicle  body 
motion.  When  the  damper  is  pushing  into  the  body,  it  is  switched  to  the  off- 
state  to  lower  the  amount  of  force  it  adds  to  the  body.  Therefore,  a  semiactive 
damper  combines  the  performance  of  a  stiff  damper  at  the  resonance 
frequency,  and  a  soft  damper  at  the  higher  frequencies,  as  shovm  in  Figure  4. 


1209 


Figure  5.  Groundhook  Damper  Configurations:  a)  optimal  groundhook 
damper  configuration,  b)  semiactive  groundhook  damper 
configuration. 

This  feature  allows  for  a  better  control  of  the  vehicle  body,  as  has  been 
discussed  in  numerous  past  studies.  The  skyhook  control  policy  in  Eq.  (4), 
however,  works  such  that  it  increases  axle  displacement,  X2,  (commonly  called 
wheel  hop).  Because  the  tire  dynamic  loading  can  be  defined  as 


DL  =  KtX2  (5) 

The  skyhook  control  actually  increases  dynamic  loading.  As  mentioned  earlier 
the  development  of  skyhook  policy  was  for  improving  ride  comfort  of  the 
vehicle,  without  losing  vehicle  handling.  Therefore,  the  dynamic  loading  of  the 
tires  was  not  a  factor  in  the  control  development. 


GROUNDHOOK  CONTROL  POLICY 

To  apply  semiactive  dampers  to  reducing  tire  dynamic  loading,  we  propose  an 
alternative  control  policy  that  can  be  implemented  in  practice  using  the  same 
hardware  needed  for  the  skyhook  policy.  To  control  the  wheel  hop,  this 
policy,  called  “groundhook,”  implies: 


X,  (;ci  -  X2 )  <  0  C  =  Con 

(6a) 

x,  (Xt  -  ^2 )  >  0  C  =  Coff 

(6b) 

As  shovm  in  Figure  5,  the  above  attempts  to  optimize  the  damping  force  on 
the  axle,  similar  to  placing  a  damper  between  the  axle  and  a  fictitious  ground 
(thus,  the  name  “groundhook”).  The  groundhook  semiactive  damper 
maximizes  the  damping  level  (i.e.,  C  =  Con)  when  the  damper  force  is  opposing 


1210 


the  motion  of  the  axle;  otherwise,  it  minimizes  the  damping  level  (i.e.,  C  = 
Coff).  The  damper  hardware  needed  to  implement  groimdhook  semiactive  is 
exactly  the  same  as 

the  skyhook  semiactive,  except  for  the  control  policy  programmed  into  the 
controller. 


SIMULATION  RESULTS 

The  model  shown  in  Figure  1  is  used  to  evaluate  the  benefits  of  groundhook 
dampers  versus  passive  and  skyhook  dampers.  A  non-linear  damper  model 
was  considered  for  the  simulations,  as  discussed  earlier.  The  road  input  was 
adjusted  such  that  the  dynamic  parameters  for  the  passive  damper  resembles 
actual  field  measurements.  Five  different  measures  were  selected  for 
comparing  the  dampers: 

•  Dynamic  Load  Coefficient  (DLC) 

•  Fourth  Power  of  the  tire  dynamic  load 

•  Sprung  mass  acceleration 

•  Rattle  Space  (relative  displacement  across  the  suspension) 

•  Axle  Displacement,  relative  to  the  road 

Dynamic  load  coefficient  and  fourth  power  of  tire  force  are  measures 
of  pavement  dynamic  loading  and  are  commonly  used  for  assessing  road 
damage.  They  are  both  considered  here  because  there  is  no  clear  consensus  on 
which  one  is  a  better  estimate  of  road  damage.  Axle  displacement,  relative  to 
the  road,  indicates  wheel  hop  and  is  directly  related  to  DLC  and  tire  force, 
therefore  it  is  yet  another  measure  of  road  damage.  Sprung  mass  acceleration 
is  a  measure  of  ride  comfort.  Our  experience,  however,  has  shown  that  for 
trucks  this  may  not  be  a  reliable  measure  of  the  vibrations  the  driver  feels  in 
the  truck.  The  relative  displacement  across  the  dampers  relates  to  the  rattle 
space,  that  is  an  important  design  parameter  in  suspension  systems, 
particularly  for  cars.  For  each  of  the  above  measures,  the  data  was  evaluated 
in  both  time  (Figures  6-7)  and  frequency  domain  (Figures  8  -  10).  In  time 
domain,  the  root  mean  square  (RMS)  and  maximum  of  the  data  for  a  five- 
second  simulation  are  compared.  In  frequency  domain,  the  transfer  function 
between  each  of  the  measures  and  road  displacement  is  plotted  vs.  frequency. 
The  frequency  plots  highlight  the  effect  of  each  damper  on  the  body  and  axle 
resonance  frequency. 

Figures  6  and  7  show  bar  charts  of  root  mean  square  (RMS)  and 
maximum  time  data,  respectively.  In  each  case  the  data  is  normalized  with 
respect  to  the  performance  of  passive  dampers  commonly  used  in  trucks. 
Therefore,  values  below  line  1 .0  can  be  interpreted  as  an  improvement  over  the 
existing  dampers.  As  Figures  6  and  7  show,  groundhook  dampers  significantly 


1211 


improve  pavement  loading,  particularly  as  related  to  the  fourth  power  of  tire 
force.  Furthermore,  the  rattle  space  is  improved  slightly  over  passive 
dampers,  indicating  that  groundhook  dampers  do  not  impose  any  additional 
burden  on  the  suspension  designers. 

One  measure  that  has  increased  due  to  groundhook  dampers  is  body 
acceleration.  As  mentioned  earlier,  in  automobiles  this  measure  is  used  as  an 
indicator  of  ride  comfort.  In  our  past  testing,  however,  we  have  found  that  for 
trucks  it  is  a  far  less  accurate  measure  of  ride  comfort.  This  is  mainly  due  to 
the  complex  dynamics  of  the  truck  frame  and  the  truck  secondary  suspension. 
A  more  accurate  measure  of  ride  comfort  is  acceleration  at  the  B-Post  (the 
post 


Tire  Axle  Body  Rattle  Space 

Dynamic  Displacment  Acceleration 

Force  ^  4 


Figure  6.  RMS  Time  Data  Normalized  with  respect  to  Passive  Damper 


Tire  Axle  Body  Rattle  Space 

Dynamic  Displacment  Acceleration 

Force  4 


Figure  7.  Max.  Time  Data  Normalized  with  respect  to  Passive  Damper 

1212 


behind  the  driver),  which  cannot  be  evaluated  from  the  single  suspension 
model  considered  here.  Nonetheless,  the  body  acceleration  is  included  for  the 
sake  of  completeness  of  data. 

The  model  shows  that  skyhook  dampers  actually  increase  the 
measures  associated  with  pavement  loading,  while  improving  body 
acceleration.  This  agrees  with  the  purpose  of  skyhook  dampers  that  are 
designed  solely  for  improving  the  compromise  between  ride  comfort  and 
vehicle  handling.  The  improvement  in  ride  comfort  occurs  at  the  expense  of 
increased  pavement  loading. 


Figure  8.  Transfer  Function  between  Axle  Displacement  and  Road  Input 


Figures  8-10  show  the  frequency  response  of  the  system  due  to  each  damper. 
In  each  figure,  the  transfer  fimction  between  one  of  the  measures  and  input 
displacement  is  plotted  vs.  frequency.  These  plots  highlight  the  impact  of 
skyhook  and  groundhook  on  the  body  and  wheel  hop  resonance,  relative  to 
existing  passive  dampers.  The  frequency  plots  indicate  that  the 


1213 


Transfer 


Frequency  (Hz) 


groundhook  dampers  reduce  axle  displacement  and  fourth  power  of  tire 
dynamic  force  at  wheel  hop  frequency.  At  body  resonance  frequency, 
groundhook  dampers  do  not  offer  any  benefits  over  passive  dampers.  The 


frequency  results  for  body  acceleration  and  rattle  space  are  similar  to  those 
discussed  earlier  for  the  time  domain  results.  The  frequency  plots  show  that 
the  skyhook  dampers  offer  benefits  over  passive  dampers  at  frequencies  close 
to  the  body  resonance  frequencies.  At  the  higher  frequencies,  associated  with 
wheel  hop,  skyhook  dampers  result  in  a  larger  peak  than  either  passive  or 
groundhook  dampers.  This  indicates  that  skyhook  dampers  are  not  suitable 
for  reducing  tire  dynamic  loading. 


CONCLUSIONS 

An  alternative  to  skyhook  control  policy  for  semiactive  dampers  was 
developed.  This  policy,  called  “groundhook,”  significantly  improves  both 
dynamic  load  coefficient  (DLC),  and  fourth  power  of  tire  dynamic  load, 
therefore  holding  a  great  promise  for  reducing  road  damage  to  heavy  trucks. 
The  dynamic  model  used  for  assessing  the  benefits  of  groundhook  dampers 
represented  a  single  suspension  system.  Although  the  results  presented  here 
show  groundhook  dampers  can  be  effective  in  reducing  tire  dynamic  loading 
and  pavement  damage,  more  complete  models  and  road  testing  are  necessary 
for  more  accurately  assessing  the  benefits. 


REFERENCES 

1.  Crosby,  M.  J.,  and  Karnopp,  D.  C.,  "The  Active  Damper,"  The  Shock 
and  Vibration  Bulletin  43,  Naval  Research  Laboratory,  Washington,  D.C., 
1973. 

2.  Karnopp,  D.  C.,  and  Crosby,  M.  J.,  "System  for  Controlling  the 
Transmission  of  Energy  Between  Spaced  Members,"  U.S.  Patent 
3,807,678,  April  1974. 

3.  Ahmadian,  M.  and  Marjoram,  R.  H.,  “Effects  of  Passive  and  Semi-active 
Suspensions  on  Body  and  Wheelhop  Control,”  Journal  of  Commercial 
Vehicles,  Vol.  98,  1989,  pp.  596-604. 

4.  Ahmadian,  M.  and  Marjoram,  R.  H.,  “On  the  Development  of  a 
Simulation  Model  for  Tractor  Semitrailer  Systems  with  Semiactive 
Suspensions,”  Proceedings  of  the  Special  Joint  Symposium  on  Advanced 
Technologies,  1989  ASME  Winter  Annual  Meeting,  San  Francisco, 
California,  December  1989  (DSC-Vol.  13). 

5.  Hedrick,  J.  K.,  "Some  Optimal  Control  Techniques  Applicable  to 
Suspension  System  Design,"  American  Society  of  Mechanical  Engineers, 
Publication  No.  73-ICT-55,  1973. 


1215 


6.  Hac,  A.,  "Suspension  Optimization  of  a  2-DOF  Vehicle  Model  Using 
Stochastic  Optimal  Control  Technique,"  Journal  of  Sound  and  Vibration, 
1985. 

7.  Thompson,  A.  G.,  "Optimal  and  Suboptimal  Linear  Active  Suspensions 
for  Road  Vehicles,"  Vehicle  System  Dynamics,  Vol.  13, 1984. 

8.  Kamopp,  D.,  Crosby,  M.  J.,  and  Harwood,  R.  A.,  "Vibration  Control 
Using  Semiactive  Force  Generators,"  American  Society  of  Mechanical 
Engineers,  Journal  of  Engineering  for  Industry,  May  1974,  pp.  619-626. 

9.  Krasnicki,  E.  J.,  "Comparison  of  Analytical  and  Experimental  Results  for 
a  Semiactive  Vibration  Isolator,"  Shock  and  Vibration  Bulletin,  Vol.  50, 
September  1980. 

10.  Chalasani,  R.M.,  "Ride  Performance  Potential  of  Active  Suspension 
Systems-Part  1:  Simplified  Analysis  Based  on  a  Quarter-Car  Model," 
proceedings  of  1986  ASME  Winter  Annual  Meeting,  Los  Angeles,  CA, 
December  1986. 

11.  Cole,  D.  J.  and  Cebon,  D.,  “Truck  Suspension  Design  to  Minimize  Road 
Damage,”  Proceedings  of  the  Institution  of  Mechanical  Engineers,  Vol. 
210,  D06894,  1996,  pp.  95-107. 

12.  Cebon,  D.,  “Assessment  of  the  Dynamic  Forces  Generated  by  Heavy 
Road  Vehicles,”  ARRB/FORS  Symposium  on  Heavy  Vehicle  Suspension 
Characteristics,  Canberra,  Australia,  1987. 

13.  Cole,  D.  J.  and  Cebon,  D.,  “Modification  of  a  Heavy  Vehicle  Suspension 
to  Reduce  Road  Damage,”  Proceedings  of  the  Institution  of  Mechanical 
Engineers,  Vol.  209,  D03594,  1995. 

14.  Yi,  K.  and  Hedrick,  J.  K.,  “Active  and  Semi- Active  Heavy  Truck 
Suspensions  to  Reduce  Pavement  Damage,”  SAE  SP-802,  paper  892486, 
1989. 


1216 


Active  Vibration  Control  of  Isotropic  Plates  Using 
Piezoelectric  Actuators 


A.  M.  Sadri",  J.  R.,  Wright*  and  A.  S.  Cherry* 

The  Manchester  School  of  Engineering,  Manchester  M13  9PL,  UK 

and 

R.  J.  Wynnes 

Sheffield  Hallam  University,  School  of  Engineering,  Sheffield,  UK 


Abstract:  Theoretical  modelling  of  the  vibration  of  plate  components  of  a 
space  structure  excited  by  piezoelectric  actuators  is  presented.  The  equations 
governing  the  dynamics  of  the  plate,  relating  the  strains  in  the  piezoelectric 
elements  to  the  strain  induced  in  the  system,  are  derived  for  isotropic  plates 
using  the  Rayleigh-Ritz  method.  The  developed  model  was  used  for  a  simply 
supported  plate.  The  results  show  that  the  model  can  predict  natural 
frequencies  and  mode  shapes  of  the  plate  very  accurately.  The  open  loop 
frequency  response  of  the  plate  when  excited  by  the  patch  of  piezoelectric 
material  was  also  obtained.  This  model  was  used  to  predict  the  closed  loop 
frequency  response  of  the  plate  for  active  vibration  control  studies  with 
suitable  location  of  sensor-actuators. 


Introduction 

Vibration  suppression  of  space  structures  is  very  important  because 
they  are  lightly  damped  due  to  the  material  used  and  the  absence  of  air 
damping.  Thus  the  modes  of  the  structure  must  be  known  very 
accurately  in  order  to  be  affected  by  the  controller  while  avoiding 
spillover.  This  problem  increases  the  difficulty  of  predicting  the 
behaviour  of  the  structure  and  consequently  it  might  cause  unexpected 
on-orbit  behaviour. 

These  difficulties  have  motivated  researchers  to  use  the 
actuation  strain  concept.  One  of  the  mechanisms  included  in  the 
actuation  strain  concept  is  the  piezoelectric  effect  whereby  the  strain 
induced  through  a  piezoelectric  actuator  is  used  to  control  the 


Research  Student,  Dynamics  &:  Control  Research  Group. 
^  Professor ,  Dynamics  &  Control  Research  Group, 

^  Former  Lecturer,  Dynamics  &  Control  Research  Group. 
^  Professor  of  Mechanical  and  Control  Engineering. 


1217 


deformation  of  the  structure  [1].  It  can  be  envisaged  that  using  this 
concept  in  conjimction  with  control  algorithms  can  enhance  the  ability 
to  suppress  modes  of  vibration  of  flexible  structures. 

Theoretical  and  experimental  results  of  the  control  of  a  flexible 
ribbed  antenna  using  piezoelectric  materials  has  been  investigated  in 
[2].  An  active  vibration  damper  for  a  cantilever  beam  using  a 
piezoelectric  polymer  has  been  designed  in  [3].  In  this  study, 
Lyapunov's  second  or  direct  method  for  distributed-parameter  systems 
was  used  to  design  control  algorithms  and  the  ability  of  the  algorithms 
was  verified  experimentally.  These  works  have  clearly  shown  the 
ability  of  piezoelectric  actuators  for  vibration  suppression.  However, 
they  have  been  limited  to  one  dimensional  systems.  Obviously,  there  is 
a  need  to  understand  the  behaviour  of  piezoelectric  actuators  in  two 
dimensional  systems  such  as  plates. 

Vibration  excitation  of  a  thin  plate  by  patches  of  piezoelectric 
material  has  been  investigated  in  [4].  Their  work  was  basically  an 
extension  of  the  one  dimensional  theory  derived  in  [1]  to  show  the 
potential  of  piezoelectric  actuators  in  two  dimensions.  In  their  studies, 
it  was  assumed  that  the  piezoelectric  actuator  doesn't  significantly 
change  the  inertia,  mass  or  effective  stiffness  of  the  plate.  This 
assumption  is  not  guaranteed  due  to  the  size,  weight  and  stiffness  of 
the  actuator.  Based  on  this  assumption,  their  model  can  not  predict  the 
natural  frequencies  of  the  plate  accurately  after  bonding  piezoelectric 
actuators.  Therefore,  it  is  essential  to  have  a  more  general  model  of  a 
plate  and  bonded  piezoelectric  actuators  with  various  boundary 
conditions.  The  model  should  be  able  to  predict  frequency  responses 
because  this  is  fundamental  to  the  understanding  of  the  behaviour  of 
the  system  for  control  design  purposes.  It  is  the  objective  of  the  current 
study  to  develop  such  a  modelling  capability. 

Previous  work  [5,  6,  7]  has  concentrated  on  the  modelling  and 
control  of  a  cantilever  beam.  The  method  used  involved  bonding 
piezoelectric  material  to  a  stiff  constraining  layer,  which  was  bonded  to 
the  beam  by  a  thin  viscoelastic  layer  in  order  to  obtain  both  active  and 
passive  damping.  Then  a  Rayleigh-Ritz  model  was  developed  and 
used  to  derive  a  linearized  control  model  so  as  to  study  different 
control  strategies.  In  the  work  described  in  this  paper,  the  method  has 
been  extended  to  the  more  complex  plate  problem.  The  paper 
introduces  a  modelling  approach  based  on  the  Rayleigh-Ritz  assumed 
mode  shape  method  to  predict  the  behaviour  of  a  thin  plate  excited  by 
a  patch  of  piezoelectric  material  bonded  to  the  surface  of  the  plate.  The 
model  includes  the  added  inertia  and  stiffness  of  the  actuator  and  has 
been  used  to  predict  the  frequency  response  of  the  plate.  Suggestions 
for  future  work  are  also  included. 


1218 


Theoretical  Modelling 

In  developing  the  Rayleigh-Ritz  model  of  a  plate  excited  by  a 
patch  of  piezoelectric  material  bonded  to  the  surface  of  the  plate,  a 
number  of  assumptions  must  be  made.  The  patch  of  piezoelectric 
material  is  assumed  to  be  perfectly  bonded  to  the  surface  of  the  plate. 
The  magnitude  of  the  strains  induced  by  the  piezoelectric  element  is  a 
linear  fimction  of  the  applied  voltage  that  can  be  expressed  by 


e:=E: 


(1) 


Here  is  the  piezoelectric  strain  constant,  is  the  piezoelectric  layer 

thickness  and  V33  is  the  applied  voltage.  The  index  31  shows  that  the 
induced  strain  in  the  '1'  direction  is  perpendicular  to  the  direction  of 
poling  '3'  and  hence  the  applied  field.  The  piezoelectric  element 
thickness  is  assumed  to  be  small  compared  to  the  plate  thickness.  The 
displacements  of  the  plate  middle  surface  are  assumed  to  be  normal  to 
it  due  to  the  bending  affects. 

Figure  1  shows  the  configuration  of  the  bonded  piezoelectric 
material  relative  to  the  surface  of  the  plate. 


w 


Figure  1.  Configuration  of  the  bonded  piezoelectric  actuator  on  the  surface  of  the 
plate. 

In  figure  1,  4  and  4  are  the  dimensions  of  the  plate,  x, ,  X2  ,  y,  and  y^ 
are  the  boundaries  of  the  piezoelectric  element  and  w,  v  and  w  are  the 
displacements  in  the  x  ,  y  and  z  direction,  respectively. 

To  derive  the  equations  of  motion  of  the  plate  based  on  the 
Rayleigh-Ritz  method,  both  the  strain  energy  U  and  kinetic  energy  T  of 
the  plate  and  the  piezoelectric  element  must  be  determined.  The  strain 


1219 


and  kinetic  energy  result  from  the  deformation  produced  by  the 
applied  strain  which  is  induced  by  exciting  the  piezoelectric  element. 
The  deformations  can  be  expressed  by  the  combination  of  the  midplane 
displacement  and  the  deformations  resulting  from  the  bending  of  the 
plate. 


Strain  Energy 

The  strain  energy  of  the  plate  and  piezoelectric  material  can  be 
calculated  by 

U  =  +T;«yY,<y)dVp  +T^T„)dVp.  (2) 

where  8  is  the  inplane  direct  strain,  a  is  the  inplane  direct  stress,  t  is 
the  inplane  shear  stress  and  y  is  shear  strain.  dV  shows  volume 
differential  and  indices  p  and  pe  refer  to  the  plate  and  piezoelectric 
actuator,  respectively.  The  strains  8^  ,  8^  and  y  can  be  shown  to  be 

For  the  Plate : 

du 

dv  d  (3) 

_du  dv  d^w 

^  dy  dx  dxdy 

For  the  Piezoelectric  actuator : 


Ev  = 


du 

a 

^~'dx~ 

^  a?* 

Sv 

a 

du  av 
«  ■  _  1  ^ 

dy  dx 

'"33=S,-H'^33 


a 


dxdy  ^ 


(4) 


where  _  refers  to  the  strains  due  to  the  deformation.  The  stresses  , 
Gy  ,  y  can  be  expressed  as 


1220 


For  the  Plate : 


E„ 

1 

0 

‘e.  ' 

1 

0 

y.vy. 

0 

0 

r.y_ 

2  . 

(5) 


For  the  Piezoelectric  actuator : 


' 

0^. 

II 

1 

0 

Sx-H^33 

1 

0 

r.y_ 

0 

0 

l-v 

2  . 

.  Is 

(6) 


where  E  is  Young  modulus  and  v  is  Poisson's  ratio  for  the  assumed 
isotropic  material.  Substituting  equations  3,  4,  5  and  6  into  2  yields  the 
strain  energy  of  the  plate  and  piezoelectric  actuator. 


^  S,  +  sj  +  1  ( 1  -  Dp)  dVp 
-  '"''33 )'  +  2Dpp(  e,  -  HV33 ){  E,  -  M.V33  )^  +  ^  1  ciVpp. 

•'Vp,  ^  _  u  —  -  tC 


(7) 


Kinetic  Energy 

To  obtain  the  kinetic  energy,  the  velocity  components  in  x,  y  and 
z  directions  are  needed.  The  velocity  components  can  be  calculated  by 
differentiating  the  displacement  components  which  are 

dw 

u—  u—  z— 
dx 

dw 

(8) 

dy 

w=w. 

Differentiating  equations  8  yields 


1221 


dw 

u=  u-  z  — 
ax 

dw 

v=  V- z — 

dy 

w 


(9) 


where  u  ,  y  and  w  are  the  velocity  components  in  the  x,  y  and  z 
directions  respectively.  Using  these  velocity  components,  the  kinetic 
energies  of  the  plate  and  piezoelectric  actuator  are  obtained  as 


dw 


P  rw^  +  (u-z^f +  (v-z^^]dV^ 


9x 


ay' 


(10) 


where  p  is  the  mass  density. 


Equation  of  Motion 

The  static  or  dynamic  response  of  the  plate  excited  by  the 
piezoelectric  actuator  can  be  calculated  by  substituting  the  strain  and 
kinetic  energy  into  Lagrange's  equation 


d  dT  dT  ^  dU 
dt  dq.  dq,"^  dq. 


(11) 


where  q^  represents  the  ith  generalised  coordinate  and  is  the  ith 
generalised  force.  As  there  are  no  external  forces  (the  force  applied  by 
5\e  piezoelectric  element  is  included  as  an  applied  strain)  or  gyroscopic 
terms  and  there  is  no  added  damping,  Lagrange's  equation  reduces  to  : 


dt  dq,  ^  dq, 


(12) 


Now  the  equation  of  motion  can  be  obtained  by  using  the  expression 
obtained  for  the  strain  and  kinetic  energy,  and  the  assumed  shape 
functions  for  flexural  and  longitudinal  motion 


1222 


u(x,y,t)~ 

v(x,y,t)={i,(x,y)y 

w(x,  y,  t)  =  {(|)(j:,  y)}^  {?(?)}. 


(13) 


Here  y  ,  ^  and  ([)  are  the  assumed  displacement  shape  and  h  ,  f  and  g 
are  generalised  coordinate  of  the  plate  response  m  x,  y  and  z 
directions.  Using  the  shape  functions  expressed  in  equations  13, 
substituting  equations  7  and  10  into  equation  12,  and  including 
Rayleigh  damping  yields  the  equation  of  motion  of  the  plate  in  the 
form 


where  M,  C  and  K  are  mass,  damping  and  stiffness  matrices  and  P  is 
the  voltage-to-force  transformation  vector.  Vector  q  represents  the 
plate  response  modal  amplitudes  and  V  is  the  applied  voltage. 

State-Space  Equations 

A  model  of  a  structure  found  via  finite  element  or  Rayliegh-Ritz 
methods  results  in  second-order  differential  equations  of  the  form 


[M]{9}+[C]{?}+[if]{?}={P}V  (15) 

Choosing  state  variables  x^_  =  q  and  =  i, ,  equation  1  may  be 

reduced  to  a  state-space  representation  as  follows  : 

q  =  x  =X2 

/  /  /  (16) 
q  =  X2=  ~M-^Kq  -  M~'Cq  +  M~'PV. 


Equations  15  can  then  be  rewritten  as 


r  0  /I 

0 

1 

1 

1 

1 

A. 

+ 

^2. 

.^2_ 

[,]  =  [/  0] 


(17) 


where  [  ],  {  }  are  ignored.  It  should  be  noted  that  the  vector  q  must  be 
multiplied  by  the  shape  fimctions  to  produce  the  actual  displacement. 


1223 


Results 

The  model  was  used  to  investigate  the  response  of  a  simply 
supported  plate.  In  order  to  maintain  symmetry  of  the  geometric 
structure  a  piezoelectric  actuator  is  assumed  to  be  bonded  to  both  the 
top  and  bottom  surfaces  of  the  plate.  So  The  symmetry  of  the  elements 
causes  no  extension  of  the  plate  midplane  and  the  plate  deforms  in 
pure  bending.  In  this  case  the  shape  functions  are  assumed  to  be  : 

\{/(x,y)  =  0 

^(x,y)  =  0  (18) 

41  ( X,  y )  =  jinf  x)sin(  ^  y) 

A- 

where  m  and  n  are  the  number  of  half  waves  in  the  x  and  y  directions. 

The  properties  of  the  plate  are  given  in  Table  1  and  its 
dimensions  are  4  =  Q38m  ,  4,  =  Q30/w  and  4=i.5876m/K.  Tables  2  and  3 
show  the  natural  frequencies  of  the  bare  plate  obtained  by  the  thin 
plate  theory  and  the  RR  model,  respectively.  Since  the  shape  functions 
used  in  this  example  express  the  exact  shape  of  the  simply  supported 
plate,  the  natural  frequencies  included  in  Tables  2  and  3  are  very  close. 

In  order  to  excite  the  plate,  a  piezoelectric  actuator  with 
configuration  x,  =  0.32  Xj  =  0.36  m,  =  0.04  OT  and  =  026  m  is  used  above 
and  below.  The  natural  frequencies  of  the  plate  after  bonding  the 
piezoelectric  actuator  to  the  surface  are  given  in  Table  4.  The  results 
show  an  increase  in  natural  frequencies,  showing  that  the  added 
stifness  is  more  important  than  the  added  inertia. 

Table  1 :  Properties  of  the  plate 

207  7870  .292 


Table  2 :  Plate  natural  frequencies  (rad  /  s) ,  Thin  Plate  Theory 


1 

2 

3 

4 

1 

437.5 

1246.0 

2593.5 

4480.0 

2 

941.4 

1749.9 

3097.4 

4983.9 

3 

1781.2 

2589.7 

3937.2 

5823.7 

4 

2957.0 

3765.5 

5113.0 

6999.5 

1224 


Table  3  :  Plate  natural  frequencies  (rad  /  s) ,  RR  Model 


/n 

1 

J 

2 

3 

4 

1 

437.5 

1245.9 

2593.2 

4479.1 

2 

941.3 

1749.7 

3097.0 

4982.8 

3 

1781.1 

2589.4 

3936.6 

5822.3 

4 

2956.7 

3764.9 

5111.9 

6997.4 

Table  4  :  Plate-Piezo  natural  frequencies  (rad  /  s) 


/n 

1 

2 

3 

4 

1 

444.0 

1257.1 

2611.3 

4502.8 

2 

957.2 

1775.6 

3182.1 

5076.7 

3 

1854.6 

2642.5 

4072.3 

6029.6 

4 

3076.6 

3933.8 

5224.6 

7277.2 

Initially,  the  piezoelectric  actuator  was  excited  by  a  constant  DC 
voltage.  The  result  of  this  action  is  shown  in  figure  2  which  shows  the 
dominant  out  of  plane  displacement  around  the  location  of  the 
piezoelectric  actuator  bonded  to  the  surface  of  the  plate.  To  show  the 
modes  of  vibration,  the  piezoelectric  actuator  was  excited  by  a  voltage 
with  frequencies  near  to  the  natural  frequencies  of  modes  (2,2)  and 
(1,3).  The  response  in  figures  3 , 4  show  that  the  piezoelectric  actuator 
excited  both  of  these  modes. 


Displacement  Distribution  (X-Y) 


0  0 


Figure  2.  Static  Displacement 


1225 


Displacement  Distribution  (X-Y) 


0  0 


Figure  3.  Vibration  of  the  plate,  mode  (2,2) 


Displacement  Distribution  (X-Y) 


0  0 


Figure  4.  Vibration  of  the  plate,  mode  (3,1) 

The  frequency  response  of  tihe  plate  at  the  centre  was  obtained 
by  exciting  the  piezoelectric  actuator  at  a  range  of  frequencies  between 
0  and  4000  rad/s.  Figure  5  shows  the  frequency  response  of  the  plate 
at  its  centre.  The  frequency  response  of  the  plate  at  y  =  0.5L^  along  the 
x-direction  is  shown  in  figure  6.  It  can  be  seen  that  the  amplitude  of 
vibration  of  some  modes  are  very  high  compared  to  that  of  the  other 
modes.  Special  attention  must  be  given  for  the  suppression  of  vibration 
of  these  modes. 


1226 


Frequency  Response  of  the  Plate 


Figure  5.  Frequency  response  of  the  plate  at  the  centre 


Frequency  Response  of  the  Plate 


4000 


Figure  6.  Frequency  response  of  the  plate  along  x-direction 

The  results  show  that  it  is  possible  to  predict  the  frequency 
response  of  a  plate  when  it  is  excited  by  a  patch  of  piezoelectric 
material.  Consequently,  a  sensor  model  can  be  also  added  to  the  model 
and  a  signal  proportional  to  velocity  fed  back  to  the  piezoelectric 
patch.  As  a  result,  the  closed  loop  frequency  response  of  the  plate  can 
be  obtained  theoretically  which  is  very  important  for  active  vibration 
control  studies.  This  also  permits  the  investigation  of  the  optimal 


1227 


location  of  the  actuators  and  the  study  of  control  algorithms  for  the 
best  possible  vibration  suppression  before  using  any  costly 
experimental  equipment. 

For  this  purpose,  two  patches  of  piezoelectric  (lOcmxScm  and 
5cmx4cm),  whose  specifications  are  listed  in  table  5,  were  bonded  to 
the  surface  of  the  plate  in  different  locations  and  then  the  plate  was 
excited  by  a  point  force  marked  by  "D"  in  the  figure  7.  In  figure  7  the 
dash  lines  are  showing  the  nodal  lines  of  a  simply  supported  plate  up 
to  mode  (3, 3). 


Table  5 :  Properties  of  the  actuator 
(mm)  EJxlO''^N/m')pJkg/m')  dJxlO'^'mfv)  ^ 

.2  6.25  7700  -180  .3 


An  actuator  is  most  effective  for  control  of  a  particular  mode  if 
the  sign  of  the  strain  due  to  the  modal  deflection  shape  is  the  same 
over  die  entire  actuator.  Consequently,  as  can  be  seen  from  figure  7, 
the  actuators  are  placed  between  the  nodal  lines  and  at  the  points  of 
maximum  curvature  in  order  to  obtain  good  damping  effect  on  the 
modes  of  interest.  Then  two  accelerometers  were  located  at  the  center 
of  the  location  of  the  actuators,  marked  by  "S"  in  figure  7,  in  order  to 
have  collocated  sensor-actuators.  The  signals  obtained  by  the 
accelerometers  are  integrated  and  fed  back  to  the  actuators  separately. 
Therefore  rate  feedback  was  used  in  this  configuration.  This  leads  to 
the  feedback  control  law 


V  =  kq 


(19) 


where  k  is  an  amplification  factor  or  feedback  gain.  Substituting 
equation  19  into  equation  17  the  closed-loop  state-space  representation 
of  the  system  can  then  be  obtained  as 


0 

-M-‘K 


_ J 

I — 

1 

M-^(C-kP)_ 

[^]=[/  0] 


(20) 


1228 


Then  the  effects  of  the  actuators  on  vibration  suppression  were 
investigated.  At  first,  only  the  actuator  near  to  the  center  of  the  plate 
was  used  to  suppress  the  vibration.  The  effect  of  this  is  shown  in  figure 
8.  As  can  be  seen,  damping  in  some  modes  are  improved  and  some 
modes  are  untouched. 

The  second  actuator  was  added  to  the  model  to  see  its  effect  on 
modes  of  vibration. 


Figure  7.  Plate  with  Bonded  Piezoelectric  Actuators 


Frequency  Response  of  the  Plate 


Figure  8.  Open  and  Closed  loop  Frequency  Response  of  the  plate 

solid  line ;  open  loop,  dash  line  :  closed  loop 


1229 


Frequency  Response  of  the  Plate 


-50 


_2qqI - 1 - 1 - 1 - 1 - 1 - 1 — — - 1 - 1 

0  500  1000  1500  2000  2500  3000  3500  4000 

Frequency  (rad/s) 

Figure  9.  Open  and  Closed  Loop  Frequency  Response  of  The  Plate 
solid  line :  open  loop,  dash  line  :  closed  loop 


The  open  and  closed  loop  frequency  response  of  the  plate  when  excited 
by  the  point  force  and  controlled  by  two  actuators  is  shown  in  figure  9. 
As  can  be  seen,  significant  vibration  suppression  was  obtained  in  both 
lower  and  higher  modes.  Also,  it  shows  that  the  place  of  actuators  was 
successfully  chosen.  This  analysis  showed  that  obtaining  reasonable 
but  not  necessarily  optimal  placement  of  actuators  in  structures  is  very 
important  in  order  to  obtain  a  high  level  of  damping  in  the  modes  of 
interest.  Obviously,  bonding  more  than  one  piezoelectric  actuator  in 
suitable  locations  helps  to  successfully  suppress  vibration  of  the  plate. 

Conclusions 

A  model  of  an  active  structure  is  fundamental  to  the  design  of 
control  strategies.  It  can  be  used  to  analyse  the  system  and  investigate 
optimal  control  strategies  without  using  costly  experimental 
equipment. 

A  Rayleigh-Ritz  model  has  been  developed  to  analyse  the  behaviour 
of  a  thin  plate  excited  by  a  patch  of  piezoelectric  material.  The  model 
has  been  used  for  a  simply  supported  plate.  It  has  been  shown  that  the 
model  can  predict  natural  frequencies  of  the  plate  alone  very 
accurately.  The  obtained  mode  shapes  also  correspond  to  the  actual 
mode  shapes.  The  frequency  response  of  the  plate  can  be  obtained  to 
show  the  suitability  of  the  model  for  control  design  studies.  This  study 
allowed  the  behaviour  of  the  system  in  open  and  closed  loop  form  for 
active  vibration  control  purposes  to  be  investigated.  Two  piezoelectric 
actuators  were  used  to  investigate  their  effectiveness  on  vibration 


1230 


suppression  of  the  plate.  The  analysis  showed  that  the  location  of  two 
actuators  was  very  important  to  increase  the  level  of  damping  in  both 
lower  and  higher  frequency  modes.  The  future  work  will  be  to  extend 
the  model  to  analyse  a  plate  with  more  than  two  patches  of 
piezoelectric  material  with  optimal  configuration,  obtained  by 
controllability  theory,  and  independent  controller  for  vibration 
suppression,  and  experimental  verification  of  the  theoretical  analysis. 


References 

1.  Crawley,  E.  F.  and  de  Luis,  J.,  Use  of  Piezoelectric  Actuators  as 
Elements  of  Intelligent  Structures,  AlAA  Journal,  Vol.  25,  No.  10, 
1987,  p.  1373. 

2.  Dosch,  J.,  Leo,  D.  and  Inman,  D.,  Modelling  and  control  for 
Vibration  Suppression  of  a  Flexible  Active  structure,  AIAA  Journal 
of  Guidance,  Control  and  Dynamics,  Vol.  18,  No.  2, 1995,  p.340. 

3.  Bailey,  T.  and  Hubbard  J.  E.  Jr.,  Distributed  Piezoelectric  Polymer 
Active  Vibration  Control  of  a  Cantilever  Beam,  AIAA  Journal  of 
Guidance,  Control  and  Dynamics,  Vol.  8,  No.  4, 1985,  p.605. 

4.  Dimitriadis,  E.  K.,  Fuller,  C.  R.,  Rogers  C.  A.,  Piezoelectric 
Actuators  for  Distributed  Vibration  Excitation  of  Thin  Plates, 
Journal  of  Vibration  and  Acoustics,  Vol.  113,  No.  1, 1991,  p.  100. 

5.  Azvine,  B.,  Tomlinson,  G.  R.  and  Wynne,  R.  J.,  Use  of  Active 
Constrained  Layer  Damping  for  Controlling  Resonant,  Journal  of 
Smart  Materials  and  Structures,  No.  4, 1995. 

6.  Rongong,  J.  A.,  Wright,  J.  R.,  Wynne,  R.  J.  and  Tomlinson,  G.  R., 
Modelling  of  a  Hybrid  Constrained  Layer/Piezoceramic  Approach 
to  Active  Damping,  Journal  of  Vibration  and  Acoustics,  To  appear. 

7.  Sadri,  A.  M.,  Wynne,  R.  J.  and  Cherry,  A.  S.,  Modelling  and 
Control  of  Active  Damping  for  Vibration  Suppression,  UKACC 
International  Conference  on  Control'  96, 2-5  September  1996. 

8.  Bathe,  K,,  Finite  Element  Procedures  in  Engineering  Analysis,  Prentice- 
Hall,  Inc.,  1982. 

9.  Blevins,  R.  D.,  Formulas  for  Natural  Frequency  and  Mode  Shapes,  Van 
Nostrand  Remhold,  NY  1979. 

10.  Thomson,  W.  T.,  Theory  of  Vibration  with  Applications,  Prentice-Hall, 
Inc.,  1988. 


1231 


Active  control  of  sound  transmission  into  a  rectangular 
enclosure  using  both  structural  and  acoustic  actuators 


S.M.  Kim  and  MJ.  Brennan 

ISVR,  University  of  Southampton,  Highfieid,  Southampton,  S017  IBJ,  UK 


ABSTRACT 

This  paper  presents  an  analytical  investigation  into  the  active  control  of  sound 
transmission  in  a  ‘weakly  coupled’  structural-acoustic  system.  The  system 
under  consideration  is  a  rectangular  enclosure  having  one  flexible  plate 
through  which  external  noise  is  transmitted.  Three  active  control  systems 
classified  by  the  type  of  actuators  are  discussed.  They  are;  i)  a  single  force 
actuator,  ii)  a  single  acoustic  piston  source,  and  iii)  simultaneous  use  of  both 
the  force  actuator  and  the  acoustic  piston  source.  For  all  three  control  systems 
the  acoustic  potential  energy  inside  the  enclosure  is  adopted  as  the  cost 
function  to  minimise,  and  perfect  knowledge  of  the  acoustic  field  is  assumed. 
The  results  obtained  demonstrate  that  a  single  point  force  actuator  is  effective 
in  controlling  well  separated  plate-controlled  modes,  whereas,  a  single 
acoustic  piston  source  is  effective  in  controlling  well  separated  cavity- 
controlled  modes  provided  the  discrete  actuators  are  properly  located.  Using 
the  hybrid  approach  with  both  structural  and  acoustic  actuators,  improved 
control  effects  on  the  plate  vibration  together  with  a  further  reduction  in 
transmitted  noise  and  reduced  control  effort  can  be  achieved.  Because  the 
acoustic  behaviour  is  governed  by  both  plate  and  cavity-controlled  modes  in  a 
‘weakly  coupled’  structural-acoustic  system,  the  hybrid  approach  is  desirable 
in  this  system. 


1.  INTRODUCTION 

Analytical  studies  of  vibro-acoustic  systems  have  been  conducted  by  many 
investigators  to  achieve  physical  insight  so  that  effective  active  control 
systems  can  be  designed.  It  is  well  established  that  a  single  point  force  actuator 
and  a  single  acoustic  piston  source  can  be  used  to  control  well  separated 
vibration  modes  in  structures  and  well  separated  acoustic  modes  in  cavities, 
respectively,  provided  that  the  actuators  are  positioned  to  excite  these 
modes  [1,2].  Active  control  is  also  applied  to  structural-acoustic  coupled 
systems  for  example,  the  control  of  sound  radiation  from  a  piate[3-6]  and  the 
sound  transmission  into  a  rectangular  enclosure[7-8].  Meirovitch  and 


1233 


Thangjitham[6],  who  discussed  the  active  control  of  sound  radiation  from  a 
plate,  concluded  that  more  control  actuators  resulted  in  better  control  effects. 
Pan  et  al[Z]  used  a  point  force  actuator  to  control  sound  transmission  into  an 
enclosure,  and  discussed  the  control  mechanism  in  terms  of  plate  and  cavity- 
controlled  modes. 

This  paper  is  concerned  with  the  active  control  of  sound  transmission  into  a 
‘weakly  coupled’  structural-acoustic  system  using  both  structural  and  acoustic 
actuators.  After  a  general  formulation  of  active  control  theory  for  structural- 
acoustic  coupled  systems,  it  is  applied  to  a  rectangular  enclosure  having  one 
flexible  plate  through  which  external  noise  is  transmitted.  Three  active  control 
systems  classified  by  the  type  of  actuators  are  compared  using  computer 
simulations.  They  are;  i)  a  single  force  actuator,  ii)  a  single  acoustic  piston 
source,  and  iii)  simultaneous  use  of  both  the  force  actuator  and  the  acoustic 
piston  source.  For  all  three  control  systems  the  acoustic  potential  energy  inside 
the  enclosure  is  adopted  as  the  cost  function  to  minimise,  and  perfect 
knowledge  of  the  acoustic  field  is  assumed.  The  effects  of  each  system  are 
discussed  and  compared,  and  a  desirable  control  system  is  suggested. 


2.  THEORY 

2.1  Assumptions  and  co-ordinate  systems 

Consider  an  arbitrary  shaped  enclosure  surrounded  by  a  flexible  structure  and  a 
acoustically  rigid  wall  as  shown  in  Figure  1.  A  plane  wave  is  assumed  to  be 
incident  on  the  flexible  structure,  and  wave  interference  outside  the  enclosure 
between  the  incident  and  radiated  waves  by  structural  vibration  is  neglected. 
Three  separate  sets  of  co-ordinates  systems  are  used;  Co-ordinate  x  is  used  for 
the  acoustic  field  in  the  cavity,  co-ordinate  y  is  used  for  the  vibration  of  the 
structure,  and  co-ordinate  r  is  used  for  the  sound  field  outside  the  enclosure. 
The  cavity  acoustic  field  and  the  flexible  structure  are  governed  by  the  linear 
Helmholtz  equation  and  the  isotropic  thin  plate  theory[9],  respectively.  The 
sign  of  the  force  distribution  function  and  normal  vibration  velocity  are  set  to 
be  positive  when  they  direct  inward  to  the  cavity  so  that  the  structural 
contribution  to  acoustic  pressure  has  the  same  sign  as  the  acoustic  source 
contribution  to  acoustic  pressure. 

Weak  coupling  rather  than  full  coupling  is  assumed  between  the  structural 
vibration  system  and  the  cavity  acoustic  system.  Thus,  the  acoustic  reaction 
force  on  the  strucural  vibration  under  structural  excitation  and  the  structural 
induced  source  effect  on  the  cavity  acoustic  field  under  acoustic  excitation  is 
neglected.  This  assumption  is  generally  accepted  when  the  enclosure  consists 
of  a  heavy  structure  and  a  big  volume  cavity.  It  is  also  assumed  that  the 
coupled  response  of  the  system  can  be  described  by  finites  summations  of  the 


1234 


uncoupled  acoustic  and  structural  modes.  The  uncoupled  modes  are  the  rigid- 
walled  acoustic  modes  of  the  cavity  and  the  in  vacuo  structural  modes  of  the 
structure.  The  acoustic  pressure  and  structural  vibration  velocity  normal  to  the 
vibrating  surface  are  chosen  to  represent  the  responses  of  the  coupled  system. 

2.2  Structural-acoustic  coupled  response 

The  acoustic  potential  energy  in  the  cavity  is  adopted  as  the  cost  function  for 
the  global  sound  control,  which  is  given  by  [2] 

where,  and  Co  respectively  denote  the  density  and  the  speed  of  sound  in  air, 
and  /?(x,ft))  is  the  sound  pressure  inside  the  enclosure. 

The  vibrational  kinetic  energy  of  the  flexible  structure,  which  will  be  used  to 
judge  the  control  effect  on  structural  vibration,  is  given  by[l] 

where,  p.,  is  the  density  of  the  plate  material,  h  is  the  thickness  of  the  plate. 

If  the  acoustic  pressure  and  the  structural  vibration  are  assumed  to  be 
described  by  a  summation  of  N  and  M  modes,  respectively,  then  the  acoustic 
pressure  at  position  x  inside  the  enclosure  and  the  structural  vibration  velocity 
at  position  y  are  given  by 

N 

P(X,(B)  = 

n=I 

M 

ni=l 

where,  the  N  length  column  vectors  ^  and  a  consist  of  the  array  of  uncoupled 
acoustic  mode  shape  functions  and  the  complex  amplitude  of  the 

acoustic  pressure  modes  a^^(co)  respectively.  Likewise  the  M  length  column 

vectors  O  and  b  consist  of  the  array  of  uncoupled  vibration  mode  shape 
functions  0n,(y)  and  the  complex  amplitude  of  the  vibration  velocity  modes 

respectively. 

The  mode  shape  functions  \f/^{x)  and  (l)Jy)  satisfy  the  orthogonal  property 
in  each  uncoupled  system,  and  can  be  normalised  as  follows. 

V  =  lwlMdV  (5) 

S,=lfyy)dS  (6) 


1235 


where,  V  and  S/  are  the  volume  of  the  enclosure  and  the  area  of  the  flexible 
structure,  respectively.  Since  mode  shape  functions  are  normalised  as  given  by 
(Eq.  5),  the  acoustic  potential  energy  can  be  written  as 

£  =_L^a“a  (7) 

'  4p„c^ 

Similarly  from  (Eq.  2)  and  (Eq.  6),  the  vibrational  kinetic  energy  can  be 
written  as 

£  =££^b«b  (8) 

2 

Where  superscript  H  denotes  the  Hermitian  transpose. 

For  the  global  control  of  sound  transmission,  it  is  required  to  have  knowledge 
of  the  complex  amplitude  of  acoustic  pressure  vector  a  for  various  excitations. 

The  complex  amplitude  of  the  n-th  acoustic  mode  under  structural  and 
acoustic  excitation  is  given  by[9, 10] 

^1.  (^)  =  4  (^)(^  (x)‘y(x,  co)dv + i//'„  (yMy,  j  (9) 

where,  s(x,Ci))  denotes  the  acoustic  source  strength  density  function  in  the 
cavity  volume  V”,  and  w(y,6))  denotes  the  normal  velocity  of  the  surrounding 
flexible  structure  on  surface  5/-.  The  two  integrals  inside  the  brackets  represent 
the  nth  acoustic  modal  source  strength  contributed  from  s(x,Q))  and  u(y,co) , 
respectively.  The  acoustic  mode  resonance  term  An(fi))  is  given  by 

A,.(co)  =  - - -  (10) 

o}--Q)-+j2C„<a„(o 

where  a)„and  C„  are  the  natural  frequency  and  damping  ratio  of  the  nth 
acoustic  mode,  respectively. 

Substituting  (Eq.  4)  into  (Eq.  9)  and  introducing  the  modal  source  strength 
q,:  =  j  ¥n  X.  0))dV ,  then  we  get 


2  /  M 

a„ (®)  =  A, (®)  (®)  +  S O',,™  •  b,„((0) 

y  V  Hi=i 


where,  C„,,„  represents  the  geometric  coupling  relationship  between  the 
uncoupled  structural  and  acoustic  mode  shape  functions  on  the  surface  of  the 
vibrating  structure  Sf  and  is  given  by[l  1] 

c,,,„  =  lvJy)<t>Jy’0>)ds  (12) 

If  we  use  L  independent  acoustic  control  sources,  can  be  written  as 

/=!  \l  /=! 


1236 


where  = - j  W ^  control  source  strength  q,^i{co) 

having  an  area  of  Scjj  is  defined  at  Xc,i. 

Thus,  the  complex  amplitude  of  acoustic  modal  pressure  vector  a  can  be 
expressed  as 

a  =  Z.(D,q,+Cb)  (14) 

2 

whereZa=  -^A. 

The  matrix  A  is  a  (NxN)  diagonal  matrix  in  which  each  (n,n)  diagonal  term 
consists  of  A„,  the  (NxM)  matrix  C  is  the  structural-acoustic  mode  shape 
coupling  matrix,  the  (NxL)  matrix  Dq  determines  coupling  between  the  L 
acoustic  source  locations  and  the  N  acoustic  modes,  the  L  length  vector  (jc  is 
the  complex  strength  vector  of  acoustic  control  sources,  and  b  is  the  complex 
vibrational  modal  amplitude  vector.  The  {NxN)  diagonal  matrix  Za  can  be 
defined  as  the  uncoupled  acoustic  modal  impedance  matrix  which  determines 
the  relationship  between  the  acoustic  source  excitation  and  the  resultant 
acoustic  pressure  in  modal  co-ordinates  of  the  uncoupled  acoustic  system. 
Generally  the  impedance  matrix  is  symmetric  but  non-diagonal  in  physical  co¬ 
ordinates,  however  the  uncoupled  modal  impedance  matrix  is  diagonal 
because  of  the  orthogonal  property  of  uncoupled  modes  in  modal  co-ordinates. 


Since  the  flexible  structure  in  Figure  1  is  assumed  to  be  governed  by  the 
isotropic  thin  plate  theory,  the  complex  vibrational  velocity  amplitude  of  the 
mth  mode  can  be  expressed  as[10] 

=  +  (15) 

p,hSf  ^ 

where,  again  p.v  is  the  density  of  the  plate  material,  h  is  the  thickness  of  the 
plate,  is  the  area  of  flexible  structure.  Inside  the  integral  /(y,ty) ,  p""\'^.(0) , 
and  p‘"'(y,ty)  denote  the  force  distribution  function,  and  the  exterior  and 
interior  sides  of  acoustic  pressure  distribution  on  the  surface  5/ ,  respectively. 
Because  of  the  sign  convention  used,  there  is  a  minus  sign  in  front  of 
(y,  (O) .  The  structural  mode  resonance  term  Bm((0)  can  be  expressed  as 


-(0^+j2^„^0),„co 


where  co^  and  Cm  natural  frequency  and  the  damping  ratio  of  m-th 


mode,  respectively.  Substituting  (Eq.  3)  into  (Eq.  15),  then  we  get 

b,niO})=——B,„ico)\  g,,,„ico)+  g^,„i(o)-'£cl„-a,(co)  (17) 

p,hSf  \  „=i  ; 


1237 


where,  L  =  L 

•  JSj-  •'■i/ 

and  Cl^  =  C,„..  ■ 

If  we  use  K  independent  point  force  actuators,  the  m-th  mode  generalised  force 
due  to  control  forces,  gc.m,  can  be  written  by 

=  XI  <P,n(y)S(y-ycjc)dsf,j,ico)  =  X^/.».t/<.-.t(®) 

k=\  ^  k=] 

(18) 

where  0,„(y)5(y-y.,t)rfS,  and  the  k-th  control  point  force  l,{co) 

JSj 

is  located  at  y^- it. 

Thus  the  modal  vibrational  amplitude  vector  b  can  be  expressed  as 

b  =  Y,(g,+Dff,-C’'a)  (19) 

where  Ys  =  — ^ —  B. 

P..hSjr 

The  matrix  B  is  a  (MxM)  diagonal  matrix  in  which  each  (m,m)  diagonal  term 
consists  of  B,„,  is  the  transpose  matrix  of  C,  the  (NxK)  matrix  Df 
determines  coupling  between  the  K  point  force  locations  and  the  M  structural 
modes,  gp  is  the  generalised  modal  force  vector  due  to  the  primary  plane  wave 
excitation,  the  K  length  vector  fc  is  the  complex  vector  of  structural  control 
point  forces,  and  a  is  the  complex  acoustic  modal  amplitude  vector.  The 
(MxM)  diagonal  matrix  Ys  can  be  defined  as  the  uncoupled  structural  modal 
mobility  matrix  which  determines  the  relationship  between  structural 
excitation  and  the  resultant  structural  velocity  response  in  modal  co-ordinates 
of  the  uncoupled  structural  system.  As  with  the  uncoupled  acoustic  impedance 
matrix  Za,  note  that  Ys  is  a  diagonal  matrix. 


From  (Eq.  14)  and  (Eq.  19),  we  get 


a  =  (l  +  Z,CY,C’^)''z,(D,q,+CY,g,+CWc) 

(20) 

b  =  (I  +  Y,C’'Z,C)''  Y,(gp  +  Dff,  -  C^Z.D^q,) 

(21) 

Since  weakly  coupling  is  assumed  i.e.  =  0  and  Y^C^Z^C  =  0  , 

then 

we  get 

a  =  Z,(D,q,+CY,g,+CY,D,f.) 

(22) 

b  =  X(g,+D,f,-C%D,q,) 

(23) 

Although  the  formulation  developed  above  covers  fully  coupled  systems,  weak 
coupling  is  assumed  hereafter  for  the  convenience  of  analysis. 


1238 


In  order  to  minimise  the  sound  transmission  into  the  cavity,  two  kinds  of 
actuators  are  used:  a  single  point  force  actuator  for  controlling  the  structural 
vibration  of  the  plate  and  a  single  rectangular  type  acoustic  piston  source  for 
controlling  the  cavity  acoustic  pressure.  The  rectangular  piston  source  is 
centred  at  (1.85,0.15,0)  with  the  area  of  0.15m  by  0.15m,  This  location  was 
chosen  because  the  sound  pressure  of  each  mode  in  a  rectangular  cavity  is  a 
maximum  at  the  corners,  and  thus  the  control  source  is  placed  away  from  the 
acoustic  nodal  planes  [2].  For  a  similar  reason,  the  point  force  actuator  is 
located  at  (9/20Li,  L2/2)  on  the  plate,  at  which  there  are  no  nodal  lines  within 
the  frequency  range  of  interest.  Table  2  shows  the  natural  frequencies  of  each 
uncoupled  systems  and  their  geometric  mode  shape  coupling  coefficients 
which  are  normalised  by  their  maximum  value.  Some  of  natural  frequencies 
which  are  not  excited  by  the  given  incident  angie((p  =  0°)  were  omitted.  The 
(m/,  m2)  and  («/,  112,  ns)  indicate  the  indices  of  the  m-th  plate  mode  and  the  n\h 
cavity  mode,  and  corresponding  the  uncoupled  natural  frequencies  of  the  plate 
and  the  cavity  are  listed.  A  total  15  structural  and  10  acoustic  modes  were  used 
for  the  analysis  under  300  Hz,  and  no  significant  difference  was  noticed  in 
simulations  with  more  modes. 


3.2  Active  minimisation  of  the  acoustic  potential  energy 

This  section  considers  an  analytical  investigation  into  the  active  control  of  the 
sound  transmission  into  the  rectangular  enclosure  in  Figure  2.  Three  active 
control  strategies  classified  by  the  type  of  actuators  are  considered.  They  are; 
i)  a  single  force  actuator,  ii)  a  single  acoustic  piston  source,  and  iii) 
simultaneous  use  of  both  the  force  actuator  and  the  acoustic  piston  source. 
Although  the  formulation  developed  in  this  paper  is  not  restricted  to  a  single 
actuator,  each  single  actuator  was  used  to  simplify  problems  so  that  the  control 
mechanisms  could  be  understood  and  effective  guidelines  for  practical 
implementation  could  be  established. 

3.2.1  control  using  a  single  force  actuator 

A  point  force  actuator  indicated  in  Figure  2  is  used  as  a  structural  actuator  and 
the  optimal  control  strength  of  the  point  force  actuator  can  be  calculated  using 
(Eq.  26).  Figure  3(a)  shows  the  acoustic  potential  energy  of  the  cavity  with 
and  without  the  control  force.  To  show  how  this  control  system  affects  the 
vibration  of  the  plate,  the  vibration  kinetic  energy  of  the  plate  obtained  from 
(Eq.  8)  is  also  plotted  in  Figure  3(b).  On  each  graph,  natural  frequencies  of  the 
plate  and  the  cavity  are  marked  and  ‘o’  at  the  frequencies,  respectively.  It 
can  be  seen  that  the  acoustic  response  of  uncontrolled  state  has  peaks  at  both 


1239 


plate  and  cavity  resonances,  and  the  vibration  response  of  uncontrolled  state  is 
governed  by  the  plate  resonances  only  because  of  ‘weak  coupling’. 

Examining  Figure  3(a,b)  it  can  be  seen  that  at  the  1st,  2nd,  4th,  and  5th  plate 
modes  corresponding  to  52  Hz,  64  Hz,  115  Hz,  and  154  Hz,  respectively  there 
is  a  large  reduction  of  the  acoustic  potential  energy.  This  is  because  the  sound 
field  at  these  frequencies  is  governed  by  the  plate  vibration  modes,  and  a 
single  structural  actuator  is  able  to  control  the  corresponding  vibration  mode  to 
minimise  sound  transmission. 

The  structural  actuator  reduces  sound  at  cavity-controlled  modes  as  well( 
especially  the  2nd  and  3rd  cavity  modes  corresponding  respectively  to  85  Hz 
and  170  Hz),  however  it  has  to  increase  plate  vibration  significantly.  It  shows 
that  minimisation  of  the  acoustic  potential  energy  does  not  always  bring  the 
reduction  of  structural  vibration,  and  vice  versa.  Since  a  cavity-controlled 
mode  is  generally  well  coupled  with  several  structural  modes,  a  single 
structural  actuator  is  not  able  to  deal  with  several  vibration  modes  because  of 
'control  spillover\A\.  This  is  the  reason  why  a  single  acoustic  piston  source 
used  in  the  next  section  was  introduced.  However,  it  is  clear  that  a  single 
point  force  actuator  is  effective  in  controlling  a  well  separated  plate-controlled 
mode  provided  the  actuator  is  not  located  close  to  the  nodal  line. 


3.2.2  control  using  a  single  piston  force  source 

A  single  acoustic  piston  source  indicated  in  Figure  2  is  used  for  controlling 
the  acoustic  sound  field  directly.  The  optimal  control  source  strength  of  the 
acoustic  piston  source  can  be  determined  using  (Eq.  26).  Figure  4  shows  the 
acoustic  potential  energy  of  the  cavity  and  the  vibrational  kinetic  energy  of  the 
plate  with  and  without  the  control  actuators. 

Since  a  plate-controlled  mode  is  generally  coupled  with  several  cavity  modes, 
the  control  effect  of  the  acoustic  source  is  not  effective  at  plate-controlled 
modes  (e.g.  52  Hz,  64  Hz,  115  Hz  etc.).  Whereas,  it  is  more  able  to  reduce 
transmitted  sound  at  the  cavity-controlled  modes  (  e.g.  0  Hz,  85  Hz,  170  Hz 
and  189  Hz)  than  the  structural  actuator.  As  can  be  noticed  from  (Eq.  22),  the 
external  incident  wave  and  force  excitation  have  the  same  sound  transmission 
mechanism,  which  is  vibrating  the  plate  and  transmitting  sound  through  the 
geometric  mode  shape  coupling  matrix  C.  Thus,  it  can  be  said  that  the 
structural  actuator  is  generally  effective  in  controlling  sound  transmission.  At 
cavity  controlled  modes,  however,  several  vibration  modes  are  coupled  with 
an  acoustic  mode.  It  means  that  a  single  acoustic  source  is  more  effective  than 
a  single  structural  actuator  since  a  single  actuator  is  generally  able  to  control 
only  one  mode.  From  the  results,  it  is  clear  that  a  single  acoustic  piston  source 
is  effective  in  controlling  well  separated  cavity-controlled  modes.  It  is 
interesting  that  there  is  not  much  difference  in  the  vibrational  kinetic  energy 


1240 


with  and  without  control  state.  It  means  that  the  acoustic  actuator  is  able  to 
reduce  sound  field  globally  without  increasing  plate  vibration. 


3.2.3  control  using  both  the  piston  source  and  the  structural  actuator 

In  this  section,  a  hybrid  approach,  simultaneous  use  of  both  the  point  force 
actuator  and  the  acoustic  piston  source,  is  applied.  The  optimal  strength  of  the 
force  actuator  and  the  piston  source  can  be  obtained  from  Eq.  (26).  Figure  5 
shows  the  acoustic  potential  energy  of  the  cavity  and  the  vibrational  kinetic 
energy  of  the  plate  with  and  without  the  control  actuators.  Even  at  the  cavity- 
controlled  modes,  it  can  be  seen  that  a  large  reduction  in  the  acoustic  potential 
energy  is  achieved  without  significantly  increasing  the  structural  vibration.  In 
general,  more  control  actuators  result  in  better  control  effects[6].  However,  the 
hybrid  approach  with  both  structural  and  acoustic  actuators  in  the  system  does 
not  merely  mean  an  increase  in  the  number  of  actuators.  As  demonstrated  in 
the  last  two  sections,  a  single  structural  actuator  is  effective  in  controlling 
well  separated  plate-controlled  modes  and  a  single  acoustic  actuator  is 
effective  for  controlling  well  separated  cavity-controlled  modes.  Since  the 
acoustic  response  is  governed  by  both  plate-controlled  and  cavity-controlled 
modes,  the  hybrid  control  approach  can  be  desirable  for  controlling  sound 
transmission  in  a  ‘weakly-coupled’  structural  acoustic  system. 

To  investigate  the  control  efforts  of  each  control  system,  the  amplitude  of  the 
force  actuator  and  the  source  strength  of  the  piston  source  are  plotted  in 
Figure  6.  There  is  a  large  decrease  of  the  force  amplitude  at  the  well 
separated  cavity-controlled  modes,  e.g.  85  Hz  and  170  Hz,  by  using  the  both 
actuators.  This  trend  can  also  be  seen  in  the  case  of  the  piston  source  strength, 
especially  at  the  1st  and  2nd  structural  natural  frequency  (52  Hz,  64  Hz).  By 
using  the  hybrid  approach,  simultaneous  use  of  both  actuators,  better  control 
effects  on  the  vibration  of  the  plate,  the  transmission  noise  reduction  and  the 
control  efforts  of  the  actuators  can  be  achieved. 


4.  Conclusion 

The  active  control  of  the  sound  transmission  into  a  ‘weakly  coupled’ 
structural-acoustic  system  has  been  considered.  The  results  obtained 
demonstrates  that  a  single  point  force  actuator  is  effective  in  controlling  well 
separated  plate-controlled  modes,  whereas,  a  single  acoustic  piston  source  is 
effective  in  controlling  well  separated  cavity-controlled  modes. 

By  using  the  hybrid  approach  with  both  structural  and  acoustic  actuators, 
improved  control  effects  on  the  plate  vibration,  further  reduction  in  sound 


1241 


transmission,  and  reduced  control  efforts  of  the  actuators  can  be  achieved. 

Since  the  acoustic  behaviour  is  governed  by  both  plate  and  cavity  resonances, 

the  hybrid  control  approach  can  be  desirable  in  controlling  sound  transmission 

in  a  ‘weakly  coupled’  structural-acoustic  system. 

References 

1.  C.R.  FULLER,  S.J.  ELLIOTT  and  P.A.  NELSONActive  control  of 
vibration,  Academic  Press  Limited,  1996 

2.  P.A.  NELSON  and  S.J.  ELLIOTT  Active  control  of  sound,  Academic 
Press  Limited,  1992 

3.  B.-T.  WANG,  C.R.  FULLER  and  K.  DIMITRIADIS  Active  control  of 
noise  transmission  through  rectangular  plates  using  multiple 
piezoelectric  or  point  force  actuators  Journal  of  the  Acoustical  Society 
of  America,  1991,  90(5),  2820-2830. 

4.  M.E.  JOHNSON  and  S.  J.  ELLIOTT  Active  control  of  sound  radiation 
using  volume  velocity  cancellation.  Journal  of  the  Acoustical  Society  of 
America,  1995,  98(4),  2174-2186. 

5.  C.R.  FULLER,  C.H.  HANSEN  and  S.D.  SNYDER  Active  control  of 
sound  radiation  from  a  vibrating  rectangular  panel  by  sound  sources 
and  vibration  inputs:  an  experimental  comparison,  Journal  of  Sound 
and  Vibration,  1991, 145(2),  195-215. 

6.  L.  MEIROVrrCH  and  S.  THANGJITHAM  Active  control  of  sound 
radiation  pressure,  Trans,  of  the  ASMS  Journal  of  Vibration  and 
Acoustics,  1990, 112,  237-244. 

7.  S.D.  SNYDER  and  N.  TANAKA  On  feedforward  active  control  of 
sound  and  vibration  using  vibration  error  signals,  Journal  of  the 
Acoustical  Society  of  America,  1993,  94(4),  2181-2193. 

8.  J.  PAN  C.H.  HANSEN  and  D.  A.  BIES  Active  control  of  noise 
transmission  through  a  panel  into  a  cavity  :  I.  analytical  study.  Journal 
of  the  Acoustical  Society  of  America,  1990,  87(5),  2098-2108. 

9.  P.M.  MORSE  and  K.U.  INGARD  Theoretical  Acoustics,  McGraw- 
Hill,  1968 

10.  E.H.  DOWELL,  G.F.  GORMAN  HI,  and  D.A.  SMITH 
Acoustoelasticity  :  general  theory,  acoustic  modes  and  forced  response 
to  sinusoidal  excitation,  including  comparisons  with  experiment. 
Journal  of  Sound  and  Vibration,  1977,  52(4),  519-542. 

11.  F.  FAHY  Sound  and  Structural  Vibration,  Radiation,  Transmission 
and  Response,  Academic  Press  Limited,  1985 


1242 


incident  plane  wave 


Figure  1  A  structural  acoustic  coupled  system  with  the  volume  V  and  its  flexible  boundary 
surface  S/. 


Figure  2  The  rectangular  enclosure  with  one  simply  supported  plate  on  the  surface  Sf  on 
which  external  plane  wave  is  incident  with  the  angles  of  (cp  =  0°)  and  (0=45°). 


Table  1  Material  properties 


Material 

Density 

Phase  speed 

Young’s 

Poisson’s 

Damping 

(kg/m^) 

(m/s) 

modulus  (N/m^) 

ratio  (v) 

ratio  (0 

Air 

1.21 

340 

- 

- 

0.01 

Steel 

7870 

- 

207x10® 

0.292 

0.01 

1243 


Table  2  The  natural  frequencies  and  geometric  mode  shape  coupling  coefficients  of  each 
uncoupled  system 


Order: 

Plate, 

1 

2 

3 

4 

■„  ,  5. 

7 

10 

■c 

’Type 

(1,1) 

(2,1):: 

0,1)  : 

(4,1) 

(5,1) 

(6,1) 

(7,1) 

Cavity^ 

'  Freq.- 

52  Hz 

64  Hz 

86Hz 

115  Hz 

154  Hz 

200  Hz 

;256Hz 

1 

(0,0,0) 

0  Hz 

0.71 

0 

0.24 

0 

0.14 

0 

O.IO 

2 

(1,0.0) 

85  Hz 

0 

0.67 

0 

0.27 

0 

0.17 

0 

3 

(2,0.0) 

170  Hz 

-0.33 

0 

0.60 

0 

0.24" 

0 

0.16 

4 

(0,0.1) 

189  Hz 

-1.00 

0 

-0.33 

0 

-0.20" 

0 

-0.14 

5 

(1,0,1) 

207.HZ 

0 

-0.94 

0 

-0.38 

0 

-0.24 

0 

6 

(2;0,1) 

254  Hz 

0.47 

0 

-0.85 

0 

-0.34 

0 

-0.22 

7 

(3,0,0) 

255  Hz 

0 

-0.40 

0 

0.57 

0 

0.22 

0 

1 oo  ISO  200  2SO 

Frequency 

(a)  the  acoustic  potential  energy  of  the  cavity 


TSO 

Frequency 


(b)  the  vibrational  kinetic  energy  of  the  plate(dB  ref .=  10  ^  J) 

Figure  3  Effects  of  minimising  the  acoustic  potential  energy  using  a  point  force  actuator  ( 
solid  line  :  without  control,  dashed  line  :  with  control  ),  where  **’  and  ‘o’  are  at  uncoupled 
plate  and  cavity  natural  frequencies,  respectively. 


301 


Frequency 

(a)  the  acoustic  potential  energy  of  the  cavity 


Figure  4  Effects  of  minimising  the  acoustic  potential  energy  using  an  acoustic  piston  source 
(  solid  line  ;  without  control,  dashed  line  :  with  control ),  where  and  ‘o’  are  at  uncoupled 
plate  and  cavity  natural  frequencies,  respectively. 


Figure  5  Effects  of  minimising  the  acoustic  potential  energy  using  both  a  point  force 
actuator  and  an  acoustic  piston  source  -  continued 


2 


Frequency 

(b)  the  vibrational  kinetic  energy  of  the  pIate(dB  ref =10‘^  J) 


Figure  5  Effects  of  minimising  the  acoustic  potential  energy  using  both  a  point  force 
actuator  and  an  acoustic  piston  source  (  solid  line  :  without  control,  dashed  line:  with  control 
),  where  and  ‘o’  are  at  uncoupled  plate  and  cavity  natural  frequencies,  respectively. 


Frequency 

(a)  the  strength  of  the  force  actuator 


Frequency 

(b)  the  strength  of  the  piston  source(  unit:  mVsec) 

Figure  6  Comparison  of  control  efforts  of  the  three  control  strategies;  using  each  actuator 
separately  (  solid  line  )  and  using  both  the  force  actuator  and  the  piston  source  (dashed  line  ) 
,  where  and  ‘o’  are  at  uncoupled  plate  and  cavity  natural  frequencies,  respectively. 


1246 


A  DISTRIBUTED  ACTUATOR  FOR  THE 
ACTIVE  CONTROL  OF  SOUND 
TRANSMISSION  THROUGH  A  PARTITION 


TJ.  Sutton,  M.E.  Johnson  and  S.J.  Elliott 

Institute  of  Sound  and  Vibration  Research 
University  of  Southampton,  Southampton  S017 IBJ 


ABSTRACT 

The  paper  considers  the  problem  of  active  control  of  soimd  transmission 
through  a  partition  using  a  single  distributed  actuator.  The  use  of  shaped, 
distributed  actuators  rather  than  point  sources  or  locally-acting  piezoceramic 
elements  offers  the  possibility  of  controlling  the  volume  velocity  of  a  plate 
without  giving  rise  to  control  spillover  and  avoids  an  increase  in  the  sound 
radiated  by  uncontrolled  structural  modes.  Specifically,  a  form  of  distributed 
piezoelectric  actuator  is  described  in  which  the  electrode  takes  the  form  of  a 
set  of  quadratic  strips  and  serves  to  apply  a  roughly  uniform  normal  force 
over  its  surface. 


INTRODUCTION 

The  strong  piezoelectric  properties  of  the  polymer  polyvinylidene  fluoride 
(PVDF)  were  discovered  in  1969  [1].  The  material  is  lightweight,  flexible, 
inexpensive  and  can  be  integrated  into  engineering  structures  for  strain 
sensing  and  to  apply  distributed  forces  and  moments  for  the  active  control  of 
vibration  and  sound  transmission.  Such  ^smarf  materials  offer  the  possibility 
of  providing  lightweight  sound-insulating  barriers  for  application  to  aircraft, 
ground-based  transport  and  in  buildings. 

Lee  [2]  has  set  out  the  underlying  theory  of  active  laminated  structures  in 
which  one  or  more  layers  of  flexible  piezoelectric  material  are  attached  to  a 
plate.  Practical  sensors  using  PVDF  material  have  been  implemented  by 
Clark  and  Fuller  [3],  Johnson  and  Elliott  [4-6],  and  others.  In  these  cases  thin 
PVDF  films  were  attached  to  the  structure  to  sense  integrated  strain  over  a 
defined  area.  In  [4]  for  example  a  distributed  sensor  was  developed  whose 
output  is  proportional  to  the  integrated  volume  velocity  over  the  surface  of  a 
plate. 


1247 


A  number  of  studies  have  been  carried  out  in  which  distributed  piezoelectric 
actuators  form  a  layer  of  a  laminated  system.  In  [7]  the  shape  of  a  distributed 
piezoelectric  actuator  was  chosen  to  be  orthogonal  to  all  but  one  of  the  natural 
modeshapes  of  the  cylindrical  shell  system  xmder  control.  Using  this 
approach  a  set  of  actuators  could  be  matched  to  the  modes  of  the  system 
under  control,  avoiding  control  spillover  {i.e.  the  excitation  of  tmcontrolled 
structural  modes). 

In  the  present  paper  a  single  shaped  PVDF  actuator  is  applied  to  a  thin  plate 
to  control  the  noise  transmission  through  it.  The  shape  of  the  actuator  is 
chosen  specifically  to  apply  an  approximately  uniform  force  to  the  plate. 
Such  an  actuator  can  be  used  to  cancel  the  total  volume  velocity  of  the  plate 
and  therefore  substantially  to  reduce  the  radiated  sound  power.  (If  volume 
velocity  is  measured  at  the  plate  surface  there  is  no  requirement  for  a  remote 
error  microphone.)  As  noted  by  Johnson  and  Elliott  [6],  the  soimd  power 
radiated  by  a  plate  which  is  small  compared  with  an  acoustic  wavelength 
depends  mainly  on  the  volume  velocity  of  the  plate.  The  simulations  in  [6] 
show  that  provided  the  plate  is  no  larger  than  half  an  acoustic  wavelength,  a 
single  actuator  used  to  cancel  volume  velocity  will  achieve  similar  results  to  a 
strategy  in  which  radiated  power  is  minimised.  It  is  possible  to  envisage  a 
large  partition  made  up  of  a  number  of  active  plate  elements  designed  on  this 
basis. 


CALCULATION  OF  NORMAL  FORCES  IN  THE  PLATE 

In  this  section  the  equation  of  motion  of  the  plate  and  attached  PVDF  layers  is 
set  out.  The  film  thickness  is  assumed  to  be  0.5  mm.  The  analysis  broadly 
follows  that  of  Dimitriadis,  Fuller  and  Rogers  [8],  but  the  individual 
piezoelectric  coefficients  and  included  separately  as  is  appropriate 

for  PVDF  and  a  sensitivity  function  is  included  to  account  for  variations  in 
electrode  shape.  In  addition  the  bending  stiffness  of  the  piezoelectric  film  is 
included  (it  is  not  negligible  as  the  whole  plate  is  covered).  The  nomenclature 
matches  that  used  by  Fuller,  Elliott  and  Nelson  [9]  but  here  the  analysis  leads 
to  the  inhomogeneous  wave  equation  for  the  plate-actuator  system. 

We  consider  an  aluminium  plate  of  thickness  Ih^  as  shown  in  Figure  1.  The 
plate  is  covered  on  its  upper  and  lower  surfaces  with  a  piezoelectric  film  of 
sensitivity  (}>(x,y)d-p  in  which  d.^  is  the  strain/ electric  field  matrix  of  the  material 
(3x6  array)  and  <^{x,y)  is  a  spatially-varying  sensitivity  function 
(0  <  (j)(x,y)  <  1 ).  The  two  piezoelectric  films  are  assumed  to  be  identical  but 
the  same  drive  voltage  is  applied  with  opposite  polarity  to  the  lower  film.  As 
a  result  of  this  antisymmetric  arrangement,  the  plate  is  subject  to  pure 
bending  with  no  straining  of  the  plate  midplane. 


1248 


In  line  with  other  similar  calculations  [2,9]  it  is  assumed  that  any  line 
perpendicular  to  the  midplane  before  deformation  will  remain  perpendicular 
to  it  when  the  plate /PVDF  assembly  is  deformed.  As  a  result,  the  strain  at 
any  point  in  the  assembly  is  proportional  to  distance  z  through  it.  (z=0  is 
defined  to  be  on  the  midplane  of  the  plate  as  indicated  in  Figure  1.)  The 
direct  and  shear  strains  throughout  the  whole  assembly  (e^,  Ey,  E^y)  are  then 
given  by  [10]: 


3^^ 


d^w 


(1-3) 


in  which  w  is  the  displacement  of  the  midplane  in  the  z  direction. 

The  corresponding  stresses  in  the  plate  (only)  follow  from  Hooke's  law  as  in 
the  standard  development  for  thin  plates: 


p  _ _ p_ 


= 


i-v: 


A  ^  p 

0''  = — — - 


2a+vj 


(4-6) 


in  which  is  the  Young's  modulus  and  Vp  is  Poisson's  ratio  for  the  plate 
material.  Stress  in  the  piezoelectric  film  follows  from  the  constitutive 
equations  for  the  material  [2].  The  direct  and  shear  stresses  for  the  upper 
piezo  film  are  designated  of  of  and  respectively: 


's/ 

K 

or"’ 

=  C  ^ 

1 - 

CO 

_^36. 

in  which  is  the  voltage  applied  across  the  actuator  fibn  (thickness  /rj  and 
4  and  are  the  strain/field  coefficients  for  the  material.  For  PVDF  d^,  = 
0,  but  it  is  included  in  the  analysis  for  completeness.  As  in  [2],  the  stiffness 
matrix  C  is  given  by: 


1249 


0 


c 


^pe  ^  pe^pe 

1  — 

^  ^  pe  ^  pe 

^  pe^pe  ^pe 

1-v'  1-v^ 

0  0 


0 

^pe 

2(l  +  v^J 


(8) 


The  stresses  in  the  lower  piezoelectric  layer  are  designated  and 

,  and  the  form  of  the  expression  is  similar  to  the  upper  layer  except  that 
the  voltage  is  applied  with  reversed  polarity: 


's/ 

^31 

V 

=  c 

+  (^{x,y) 

a 

Summing  moments  about  the  x  and  y  axes  for  a  small  element  dxby  of  the 
plate  yields  the  moment  per  unit  length  about  the  y-axis  and  My  about 
the  x-axis;  also  the  corresponding  twisting  moments  per  imit  length,  My^  and 


M^=jc!’’zdz+ 

-III,  ~K-K  ^‘b 

(10) 

h,,  -III,  ‘‘b+lh, 

My  =  j  a^yZdz  +  J  +  J  cf'^zdz 

-III,  -hh-K  h 

(11) 

h,,  -III,  l>h+lh, 

=  |<t;,z*+  ja^fhdz 

(12) 

-III,  -K-K  h 


and  My^=M,y. 


The  vertical  acceleration  at  each  point  of  the  plate  d^w/dt^  is  obtained  by 
taking  moments  about  the  x  and  y  axes  for  a  small  element  6x6y  and 
resolving  vertical  forces  as  in  standard  thin  plate  theory.  If  the  plate  is  acted 
on  by  some  external  force  per  unit  area  p(x,y)f(t)  then  the  vertical  motion  of 

the  plate  is  described  by: 


dx^  dxdy  dy^  dt^ 


=  -p(x,y)f(t) 


(13) 


1250 


in  which  m  is  the  mass/ area  of  the  plate-film  assembly. 


The  equation  of  motion  of  the  plate  complete  with  attached  piezoelectric  film 
is  obtained  by  combining  the  above  equations.  For  convenience  the  following 
constants  are  defined; 


n  _ 

''  3(1 


(bending  stiffness  of  plate,  thickness  2/i,)  (14) 


3(1 -V^) 


(bending  stiffness  due  to  PVDF)  (15) 


+2/1,) 


The  equation  of  motion  of  the  plate  assembly  including  upper  and  lower 
piezoelectric  layers  is  then  given  by: 


3^(l)(^,y) 


3>(j:,y) 


The  left-hand  side  of  this  equation  determines  the  free  response  of  the  plate- 
film  assembly  and  is  recognised  as  the  standard  form  for  a  thin  plate.  The 
first  term  on  the  right-hand  side  is  the  assumed  externally  applied  normal 
force  per  imit  area.  The  second  term  on  the  right-hand  side  gives  the  effective 
normal  force  per  unit  area  applied  to  the  plate  due  to  the  two  piezoelectric 
films  driven  by  a  voltage  .  It  is  clear  that  this  force  depends  on  the 
spatially-varying  sensitivity  which  has  been  assumed  for  the  piezoelectric 
material. 

An  examination  of  Eqns.  (16)  and  (17)  shows  that  the  normal  force  applied  by 
the  piezoelectric  film  depends  on  the  sum  of  the  plate  thickness  and  the 
thickness  of  one  of  the  film  layers.  If  the  film  is  much  thinner  than  the  plate 
(h^ «  hy)  then  the  applied  force  becomes  independent  of  the  film  thickness  and 
depends  only  on  the  plate  thickness,  the  applied  voltage,  the  electrode  pattern 
and  the  material  constants.  Eqn.  (17)  also  shows  that  the  normal  force  is 
applied  locally  at  all  points  on  the  plate.  No  integration  is  involved,  and  so  in 
contrast  with  a  volume  velocity  sensor  designed  using  quadratic  strips  [5],  the 
force  does  not  depend  on  the  plate  boundary  conditions  in  any  way. 
Furthermore,  there  is  no  need  to  use  two  films  oriented  at  90  degrees  to  cancel 

1251 


the  cross-sensitivity  It  is  also  worth  noting  that  no  assumption  has  been 
made  about  the  modeshapes  on  the  plate. 

We  can  create  a  uniform  force  actuator  by  choosing: 

^  =  constant,  and  (18) 

Bx 

(19) 

This  can  approximately  be  achieved  by  depositing  electrodes  in  the  form  of 
narrow  strips  whose  width  varies  quadratically  in  the  x-direction.  (Note  that 
the  x-direction  is  defined  as  the  direction  of  rolling  of  the  PVDF  material,  Le. 
the  direction  of  for  maximum  sensitivity.)  The  form  of  the  electrodes  is  as 
shown  in  Figure  2.  With  this  pattern  the  sensitivity  function  takes  the  form: 

(t)(x,>')  =  (20) 

where  is  the  length  of  the  strip.  Thus  (t)(-^>^)  -  ^  at  x  =  0  and  x  =  (no 
electrode),  while  (|)(x,  31)  =  1  halfway  along  at  x  =  LJ2  (electrode  fully  covers 
the  film). 


APPLICATION  TO  A  THIN  ALUMINIUM  PLATE 

If  a  plane  wave  of  sound  pressure  level  94  dB  (say)  is  normally  incident  on  a 
hard  surface,  it  will  exert  a  pressure  of  2  Pa  rms  on  that  surface.  If  this 
incident  pressure  is  counterbalanced  by  a  uniform  force  actuator  applied  to  a 
plate,  then  the  plate  could  in  principle  be  brought  to  rest.  Thus  for  active 
control  of  everyday  noise  levels  the  uniform  force  actuator  will  need  to  be 
able  to  generate  a  normal  force /area  of  a  few  pascal  over  the  surface  of  the 
plate.  (When  the  incident  wave  impinges  on  the  plate  at  an  oblique  angle, 
many  natural  modes  of  the  plate  will  be  excited  and  it  will  not  be  possible  to 
bring  it  perfectly  to  rest  with  a  single  actuator;  however  it  will  remain 
possible  to  cancel  the  plate  volume  velocity  as  explained  earlier.) 

By  way  of  example  an  aluminium  plate  of  thickness  1  mm  will  be  assumed, 
with  a  free  surface  measuring  300  x  400  mm.  Attached  to  each  side  is  a  PVDF 
film  of  thickness  0.5  mm.  One  electrode  of  each  panel  would  be  masked  to 
give  quadratic  strips  of  length  300  mm  as  shown  in  Figure  2.  (The  width  of  the 
strips  is  unimportant,  but  should  be  significantly  smaller  than  the  structural 
wavelength  of  modes  of  interest  on  the  plate.)  In  this  case  it  turns  out  that 

=  6.64  for  the  plate,  and 


1252 


Dp,  =  1.27  for  the  PVDF  film. 

The  piezoelectric  constants  for  the  film  are  typically 

4  =  23  X  m/V  and 

4  =  3  X  lO’"'  m/V,  giving 

Cp,  =  3.28x10^ 

Finally  the  double  derivative  of  the  sensitivity  function  turns  out  to  be 

=  88.9  (300  mm  strip  length) 

dx^ 

The  bending  stiffness  of  the  1  mm  aluminium  plate  is  increased  by  20%  due  to 
the  addition  of  two  layers  of  PVDF  film  of  thickness  0.5  mm  each.  The  force 
per  unit  area  due  to  the  actuator  is  obtained  from  Eqn.  (17): 

force/area  =  4,(^3, +Vp,4)|^'^3 

=  6.96x10’^  V3  Pa 

Thus  1000  volt  rms  would  yield  a  tmiform  force/area  of  close  to  7  Pa.  This  is 
not  an  impracticable  voltage  level,  but  previous  experience  at  ISVR  suggests 
that  care  would  need  to  be  taken  to  avoid  electrical  breakdown  through  the 
air  between  electrodes,  or  over  damp  surfaces. 


ACTIVE  CONTROL  OF  SOUND  TRANSMISSION 

In  reference  [6]  Johnson  and  Elliott  have  presented  simulations  of  the  active 
control  of  harmonic  sound  transmitted  through  a  plate  using  a  uniform  force 
actuator.  Their  actuator  might  be  realised  along  the  lines  described  in  this 
paper.  In  the  simulatioi\s  presented  in  [6]  the  uniform  force  actuator  is  used 
with  a  matched  volume  velocity  sensor  having  the  same  electrode  shape  [4]. 
The  advantage  of  this  configuration  is  that  the  actuator  can  be  used  to  drive 
the  net  volume  velocity  of  the  plate  to  zero  without  exciting  high  order 
structural  modes  in  the  process  (control  spillover).  Simulations  of  a 
300x380x1  mm  aluminium  plate  showed  that  reductions  in  transmitted  sound 
power  of  around  10  dB  were  achievable  in  principle  up  to  600  Hz  using  this 
matched  actuator-sensor  arrangement. 

A  further  advantage  of  the  distributed  matched  actuator-sensor  pair  is  that 
the  secondary  path  through  the  plate  (for  active  control)  is  minimum  phase 
[6],  giving  good  stability  characteristics  if  a  feedback  control  loop  is 
implemented  to  control  random  incident  soimd  for  which  no  reference  signal 
is  available. 


1253 


CONCLUSION 


A  design  of  distributed  piezoelectric  actuator  has  been  presented  which 
generates  a  roughly  uniform  force  over  the  surface  of  a  plate.  An  example 
calculation  shows  that  the  design  is  capable  of  controlling  realistic  soimd 
pressure  levels.  When  used  in  combination  with  a  matched  volume  velocity 
sensor,  the  actuator-sensor  pair  will  have  minimum-phase  characteristics  and 
will  offer  the  possiblity  of  feedback  control  in  which  neither  a  reference  signal 
nor  a  remote  error  sensor  will  be  required. 


ACKNOWLEDGEMENT 

The  financial  support  of  the  European  Community  under  the  Framework  IV 
programme  is  gratefully  acknowledged.  (Project  reference:  BRPR-CT96-0154) 


REFERENCES 

[1]  G.M.  Sessler  (1981)  JAcoust  Soc  Am  70(6)  Dec  1981 1596-1608 
Piezoelectricity  in  polyvinylidene  fluoride 

[2]  C.K.  Lee  (1990)  JAcoust  Soc  Am  87(3)  Mar  1990  1144-1158  Theory  of 
laminated  piezoelectric  plates  for  the  design  of  distributed  sensors/ actuators. 
Part  I:  Governing  equations  and  reciprocal  relationships 

[3]  R.L.  Clark  and  C.R.  Fuller  (1992)  JAcoust  Soc  Am  91(6)  June  1992  3321-3329 
Modal  sensing  of  efficient  acoustic  radiators  with  polyvinylidene  fluoride 
distributed  sensors  in  active  structural  acoustic  control  approaches 

[4]  M.E.  Johnson,  S.J.  Elliott  and  J.A.  Rex  (1993)  ISVK  Technical  Memorandum 
723.  Volume  Velocity  Sensors  for  Active  Control  of  Acoustic  Radiation 

[5]  M.E.  Johnson  and  S.J.  Elliott  (1995)  Proceedings  of  the  Conference  on  Smart 
Structures  and  Materials  27  Feb-3  Mar  1995,  San  Diego,  Calif.  SPIE  Vol  2443. 
Experiments  on  the  active  control  of  sound  radiation  using  a  volume  velocity 
sensor 

[6]  M.E.  Johnson  and  S.J.  Elliott  (1995)  JAcoust  Soc  Am  98(4)  Oct  1995  2174- 
2186.  Active  control  of  sound  radiation  using  volume  velocity  cancellation 

[7]  H.S.  Tzou,  J.P.  Zhong  and  J.J.  Hollkamp  (1994)  Journal  of  Sound  and 
Vibration  177(3)  363-378  Spatially  distributed  orthogonal  piezoelectric  shell 
actuators:  theory  and  applications 

[8]  E.K.  Dimitriadis,  C.R.  Fuller  and  C.A.  Rogers  (1991)  Transactions  of  the 
ASME,  Journal  of  Vibration  and  Acoustics  113  100-107  Piezoelectric  actuators  for 
distributed  vibration  excitation  of  thin  plates 

[9]  C.R.  Fuller,  S.J.  Elliott  and  P.A.  Nelson  (1996)  Active  Control  of  Vibration. 
Academic  Press,  London. 

[10]  G.B.  Warburton  (1976)  The  Dynamical  Behaviour  of  Structures,  2nd  Edition. 
Pergamon  Press,  Oxford. 


1254 


layers  of 
PVDF  film 


Figure  1 :  Schematic  diagram  of  thin  plate 
covered  on  both  sides  with  a  layer  of  PVDF  film 


Figure  2:  Electrode  pattern  of  quadratic  strips  for 
uniform-force  actuator 


1255 


1256 


CONTROL  OF  SOUND  RADIATION  FROM  A  FLUID-LOADED  PLATE 
USING  ACTIVE  CONSTRAINING  LAYER  DAMPING 

J.  Ro,  A.  Al-Ali  and  A.  Baz 

Mechanical  Engineering  Department 
The  Catholic  University  of  America 
Washington  D.  C.  20064 

Abstract 

Sound  radiation  from  a  vibrating  flat  plate,  with  one  side  subjected  to 
fluid-loading,  is  controlled  using  patches  of  Active  Constrained  Layer  Damping 
(ACLD).  The  fluid-structure-controller  interaction  is  modeled  using  the  finite 
element  method.  The  damping  characteristics  of  the  ACLD/plate/fluid  system  are 
determined  and  compared  with  the  damping  characteristics  of  plate/fluid  system 
controlled  with  conventional  Active  Control  (AC)  and/or  Passive  Constrained 
Layer  Damping  (PCLD)  treatments.  Such  comparisons  are  essential  in 
quantifying  the  individual  contribution  of  the  active  and  passive  damping 
components  to  the  overall  damping  characteristics,  when  each  operates  separately 
and  when  both  are  combined  to  interact  in  unison  as  in  the  ACLD  treatments. 

I.  INTRODUCTION 

When  a  structure  is  in  contact  with  or  immersed  in  a  fluid,  its  vibration 
energy  radiates  into  the  fluid  domain.  As  a  result,  there  is  an  observable  increase 
in  the  kinetic  energy  of  the  structure  due  to  the  fluid  loading.  Because  of  this 
kinetic  energy  increase,  the  natural  frequencies  of  structures  which  are  subjected 
to  fluid-loading  decrease  significantly  compared  to  the  natural  frequencies  of 
structures  in  vacuo.  Therefore,  through  understanding  of  the  interaction  between 
the  elastic  plate  structures  and  the  fluid  loading  has  been  essential  to  the  effective 
design  of  complex  structures  like  ships  and  submarine  hulls.  Lindholm  et  al.  [1] 
used  a  chordwise  hydrodynamic  strip  theory  approach  to  study  the  added  mass 
factor  for  cantilever  rectangular  plates  vibrating  in  still  water.  Fu  et  al  [2]  studied 
the  dry  and  wet  dynamic  characteristics  of  vertical  and  horizontal  cantilever 
square  plates  immersed  in  fluid  using  linear  hydroelasticity  theory.  Ettouney  et 
al  [3]  studied  the  dynamics  of  submerged  structures  using  expansion  vectors, 
called  wet  modes  which  are  finite  series  of  complex  eigenvectors  of  the  fluid- 
structure  system.  Recently  Kwak  [4]  presented  an  approximate  formula  to 
estimate  the  natural  frequencies  in  water  from  the  natural  frequencies  in  vacuo. 

When  the  structure  and  the  fluid  domains  become  rather  complex, 
solutions  of  fluid-structure  coupled  system  can  be  obtained  by  finite  element 


1257 


methods.  Marcus  [5],  Chowdhury  [6],  Muthuveerappan  et  al.  [7]  and  Rao  et  al 
[8]  have  successfully  implemented  the  finite  element  method  to  predict  the 
dynamic  characteristics  of  elastic  plates  in  water.  Everstine  [9]  used  both  finite 
and  boundary  element  methods  to  calculate  the  added  mass  matrices  of  fiilly- 
coupled  fluid-structure  systems. 

The  above  investigations  formed  the  bases  necessary  to  devising  passive 
and  active  means  for  controlling  the  vibration  of  as  well  as  the  sound  radiation 
from  fluid-loaded  plates.  Passive  Constrained  Layer  Damping  (PCLD)  treatments 
have  been  used  extensively  and  have  proven  to  be  effective  in  suppressing 
structural  vibration  as  reported,  for  example,  by  Jones  and  Salerno  [10],  Sandman 
[11]  and  Dubbelday  [12].  Recently,  Gu  and  Fuller  [13]  used  feed-forward  control 
algorithm  which  relied  in  its  operation  on  point  forces  to  actively  control  the 
sound  radiation  from  a  simply-supported  rectangular  fluid-loaded  plate. 

In  the  present  study,  the  new  class  of  Active  Constrained  Layer  Damping 
(ACLD)  treatment  is  utilized  as  a  viable  alternative  to  the  conventional  PCLD 
treatment  and  Active  Constrained  (AC)  with  PCLD  treatment  (AC/PCLD).  The 
ACLD  treatment  proposed  combines  the  attractive  attributes  of  both  active  and 
passive  damping  in  order  to  provide  high  energy  dissipation-to-weight 
characteristics  as  compared  to  the  PCLD  treatment.  Such  surface  treatment  has 
been  successfully  employed  to  control  the  vibration  of  various  structural  members 
as  reported,  for  example,  by  Shen  [14]  and  Baz  and  Ro  [15].  In  this  paper,  the  use 
of  the  ACLD  is  extended  to  the  control  of  sound  radiation  from  fluid-loaded 
plates.  Finite  element  modeling  of  the  dynamics  and  sound  radiation  of  fluid- 
loaded  plates  is  developed  and  validated  experimentally.  Particular  focus  is 
placed  on  demonstrating  the  effectiveness  of  the  ACLD  treatment  in  suppressing 
the  structural  vibration  and  attenuating  the  sound  radiation  as  compared  to 
conventional  PCLD  and  AC/PCLD. 

This  paper  is  organized  in  five  sections.  In  Section  1,  a  brief  introduction 
is  given.  In  Section  2.,  the  concepts  of  the  PCLD,  ACLD  and  AC/PCLD 
treatments  are  presented.  In  Section  3,  the  dynamical  and  fluid  finite  element 
models  are  developed  to  describe  the  interaction  between  the  plate,  ACLD  and  the 
contacting  fluid.  Experimental  validation  of  the  models  are  given  in  Section  4. 
Comparisons  between  the  theoretical  and  experimental  performance  are  also 
presented  in  Section  4  for  different  active  and  passive  damping  treatments. 
Section  5,  summarizes  the  conclusions  of  the  present  study. 

2.  CONCEPTS  OF  PCLD,  ACLD  AND  AC/PCLD  TREATMENTS 

Figures  (1-a),  (1-b)  and  (1-c)  show  schematic  drawings  of  the  PCLD, 
ACLD  and  AC/PCLD  treatments  respectively.  In  Figure  (1-a),  the  plate  is  treated 


1258 


Figure  (1)  -  Schematic  drawing  of  different  surface  treatments  (a)  PCLD,  (b) 
ACLD  and  (c)  AC/PCLD. 


with  a  viscoelastic  layer  which  is  bonded  directly  to  the  plate.  The  outer  surface 
of  the  viscoelastic  layer  is  constrained  by  an  inactive  piezo-electric  layer  in  order 
to  generate  shear  strain  y,  which  results  in  dissipation  of  the  vibrational  energy  of 
the  plate.  Activating  the  constraining  layer  electrically,  generates  a  control  force 
Fp  by  virtue  of  the  piezo-electric  effect  as  shown  in  Figure  (l-b)  for  the  ACLD 
treatment  .  Such  control  action  increases  the  shear  strain  to  yj  which  in  turn 
enhances  the  energy  dissipation  characteristics  of  the  treatment.  Also,  a  restoring 
moment  Mp=d2Fp  is  developed  which  attempts  bring  the  plate  back  to  its 
undeformed  position.  In  the  case  of  AC/PCLD  treatment,  shown  in  Figure  (l-c), 
two  piezo- films  are  used.  One  film  is  active  and  is  bonded  directly  to  the  plate  to 
control  its  vibration  by  generating  active  control  (AC)  force  Fp  and  moment 
Mp^djFp.  The  other  film  is  inactive  and  used  to  restrain  the  motion  of  the 


1259 


viscoelastic  layer  in  a  manner  similar  to  the  PCLD  treatment  of  Figure  (1-a).  In 
this  way,  the  AC  action  operates  separately  from  the  PCLD  action.  This  is  unlike 
the  ACLD  configuration  where  the  active  and  passive  control  actions  operate  in 
unison.  Note  that  in  the  ACLD  configuration,  larger  shear  strains  are  obtained 
hence  larger  energy  dissipation  is  achieved.  Furthermore,  larger  restoring 
moments  are  generated  in  the  ACLD  treatments  as  compared  to  the  AC/PCLD 
treatments  as  the  moment  arm  d2  in  the  former  case  is  larger  than  the  moment  arm 
d3  of  the  latter  case.  This  results  in  effective  damping  of  the  structural  vibrations 
and  consequently  effective  attenuation  of  sound  radiation  can  be  obtained. 


3.  FINITE  ELEMENT  MODELING 

3.1  Overview 

A  finite  element  model  is  presented  in  this  section,  to  describe  the 
behavior  of  fluid-loaded  thin  plates  which  are  treated  with  ACLD,  PCLD  and 
AC/PCLD  treatments. 

3.2  Finite  Element  Model  of  Treated  Plates 


(b)  (c) 

Figure  (2)  -  Schematic  drawing  of  plate  with  ACLD/AC/PCLD  patches. 


Figure  (2)  shows  a  schematic  drawing  of  the  ACLD  and  AC/PCLD 
treatments  of  the  sandwiched  plate  which  is  divided  into  N  finite  elements.  It  is 
assumed  that  the  shear  strains  in  the  piezo-electric  layers  and  in  the  base  plate  are 
negligible.  The  transverse  displacement  w  of  all  points  on  any  cross  section  of  the 
sandwiched  plate  are  considered  to  be  the  same.  The  damping  layers  are  assumed 
to  be  linearly  viscoelastic  with  their  constitutive  equations  described  by  the 
complex  shear  modulus  approach  such  that  G=G  (1+T|i).  In  addition,  the  bottom 
piezo-electric  layer  (AC)  and  the  base  plate  are  considered  to  be  perfectly  bonded 
together  and  so  are  the  viscoelastic  layer  and  the  top  piezo-electric  layer. 


The  treated  plate  elements  considered  are  two-dimensional  elements 
bounded  by  four  nodal  points.  Each  node  has  seven  degrees  of  fireedom  to 
describe  the  longitudinal  displacements  u,  and  v,  of  the  constraining  layer,  U3  and 
V3  of  the  base  plate,  the  transverse  displacement  w  and  the  slopes  w  and  W  y  of 
the  deflection  line.  The  deflection  vector  {5}  can  be  written  as: 

{5}  =  {u„v„U3,  V3,W,  W  ^W  y}"" 

=  [{n,}  {N3}  {Nj  {n.}  {n,}  {N,}^ 

where  {5"}  is  the  nodal  deflection  vector,  {Nj},  {Nj},  {N3},  {N4},  {N5},  {N5} ,,, 
and  {N5}  y  are  the  spatial  interpolating  vectors  corresponding  to  u„  v„  U3,  V3,  w, 
w^,  and  Wy  respectively.  Subscripts  ,x  and  ,y  denote  spatial  derivatives  with 
respect  to  x  and  y. 

Consider  the  following  energy  functional  ITp  for  the  treated  plate/fluid 

system: 

np  =  I(u-TK+w.-w^+wJdv,  (2) 

where  U  is  the  strain  energy,  T^  is  the  kinetic  energy,  is  work  done  by  external 
forces,  Wp  is  work  done  by  the  back  pressure  inside  the  fluid  domain,  is  work 
done  by  the  control  forces  and  moments  and  V  is  the  volume  of  the  plate.  These 
energies  are  expressed  as  follows 


I T,dv  =X  ^  1 1„  (*'  =  5{®T['^p]{®')  ’ 

|w,dV=j{5'f{F}, 


=  5'  Kj  5' 


“  V.l^dxdy 

„[Ui  hi  ‘J  “  ax^  Ui  ‘  hi  j  “  Sy^J 

where  i=l  for  ACLD  control  or  i=3  for  AC  control  (6) 


and  I  WpdV  ={5'}’"[n  ]{p'}. 


where  {p®}  is  the  nodal  pressure  vector  of  the  fluid  element.  In  the  above 
equation  []^],  [Mp],  {F},  [KJ  and  [Q]  are  the  plate  stiffness  matrix,  mass  matrix, 
external  forces  vector,  piezo-electric  forces  and  moments  matrix  and  plate/fluid 
coupling  matrix  as  given  in  the  appendix.  In  equation  (6),  d3i  32  are  the  piezo¬ 
strain  constants  in  directions  1  and  2  due  to  voltage  applied  in  direction  3.  The 
voltage  is  generated  by  feeding  back  the  derivative  of  the  displacement  5  at 

critical  nodes  such  that  j  where  is  the  derivative  feedback  gain 

matrix  and  C  is  the  measurement  matrix  defining  the  location  of  sensors. 

Minimizing  the  plate  energy  fimctional  using  classical  variational  methods 

such  that  |anp/a{6®}j  =  0  leads  to  the  following  finite  element  equation: 

{[K]-»lM.]){5'}-[n]{p'}  =  {F}  (8) 

where  co  is  the  frequency  and  [K]  =  [Kp]  +  [K^.]  is  overall  stiffriess  matrix. 

3.3  Finite  Element  Model  of  the  Fluid 

The  fluid  model  uses  solid  rectangular  tri-linear  elements  to  calculate  the 
sound  pressure  distribution  inside  the  fluid  domain  and  the  associated  structural 
coupling  effects.  The  fluid  domain  is  divided  into  fluid  elements.  Each  of 


1262 


these  elements  has  eight  nodes  with  one  degree  of  jfreedom  per  node.  The 
pressure  vector  is  expressed  by  p  =  [Nf]{p®}  and  [NJ  is  pressure  shape  function 
and  {p®}  is  nodal  pressure  vector. 

Considering  the  following  functional  Elf  of  fluid  domain  Craggs  [16] 

where  [Kf]  and  [MJ  are  the  fluid  stiffiiess  and  mass  matrices  as  given  in  the 
appendix.  Minimization  of  equation  (9)  such  that  {OTf/^lp^}}  =  0  yields  the 
fluid  dynamics  as  coupled  with  the  structural  vibration: 

([K,]-o)^[M,]){p'}-<a^[nf{5'}  =  {0}  (10) 

The  boundary  conditions  involved  are  of  the  form 

ap/an  =  0,  at  a  rigid  boundary 


a  p  /  a  n  =  -pp  9^5  /  9 , 
and  p  =  0. 


at  a  vibrating  boundary 
at  a  free  surface 


where  pf  is  fluid  density. 

3.4.  Solutions  of  the  Coupled  Plate/Fluid  System 

Combining  equations  (8)  and  (10)  gives 

■[K]-co^[m,]  -[fi]  Ip'l.pl  (11) 

[Kr]-CD^[M,]J  Lp'J  k 

At  low  frequencies,  the  fluid  pressure  is  in  phase  with  the  structural 
acceleration,  i.e.  the  fluid  appears  to  the  structure  like  an  added  mass.  However, 
as  the  frequency  increases  the  added  mass  effect  diminishes  and  the  damping 
effect,  i.e.  the  pressure  proportional  to  velocity,  increases.  For  an  incompressible 
fluid,  the  speed  of  sound  c  approaches  oo,  thus  the  mass  matrix  of  the  fluid  [MJ 
vanishes,  and  equation  (11)  can  be  simplified  to 

■[K]-cd=[m,]  -[fl]]  rs']  pi  (12) 

_  [KfiJkJ  k 

If  the  fluid-structure  coupled  system  has  free  boundary  surface,  then  [KJ  is 
non-singular  [Everstin,  1991]  and  the  nodal  pressure  vector  {p®}  can  be  eliminated 
from  equation  (12)  as  follows: 

{p'}=-o.^[K,nnr{5'}  (13) 


1263 


Hence,  equation  (12)  yields 


([K]-ffl^(K]+[Mj)){8')  =  {F)  (14) 

where  [MJ  is  added  virtual  mass  matrix  defined  by  [Muthuveerappan,  1979] 

[Mj  =  [a][K,]-'[£2f  (15) 

Equation  (14)  only  involves  the  unknown  nodal  deflection  vector  {5®}  of  the 
structure.  When  {F}=0,  equation  (14)  becomes  an  eigenvalue  problem,  the 
solution  of  which  yields  the  eigenvalues  and  eigenvectors.  The  nodal  pressures 
can  then  be  obtained  from  equation  (13)  when  the  nodal  displacements  are 
determined  for  any  particular  loading  on  plate. 

4.  PERFORMANCE  OF  PARTIALLY  TREATED  PLATES  WITH 
FLUID  LOADING 

In  this  section,  comparisons  are  presented  between  the  numerical 
predictions  and  experimental  results  of  the  natural  frequencies  and  damping  ratios 
of  a  fluid-loaded  plate  treated  with  ACLD,  PCLD  and  AC/PCLD.  The  effect  of 
the  Active  Control,  Passive  Constrained  Layer  Damping  and  Active  Constrained 
Layer  Damping  on  the  resonant  frequency,  damping  ratio,  attenuation  of  vibration 
amplitude  and  sound  radiation  are  investigated  experimentally.  The  vibration  and 
sound  radiation  attenuation  characteristics  of  the  fluid-structure  coupled  system 
are  determined  when  the  plate  is  excited  acoustically  with  broadband  frequency 
excitation  while  the  piezo-electric  layers  are  controlled  with  various  control  gains. 
The  experimental  results  are  compared  with  the  theoretical  predictions. 

4.1  Experimental  Set-up 

Figures  (3-a)  and  (3-b)  show  a  schematic  drawing  and  finite  element  mesh 
of  the  experimental  set-up  along  with  the  boundary  conditions  used  to  describe 
the  fluid-structure  system.  The  finite  element  mesh  includes:  24  plate-elements 
and  560  fluid-elements.  The  coupled  system  has  a  total  of  815  active  degree  of 
freedoms.  The  aluminum  base  plate  is  0.3m  long,  0.2m  wide  and  0.4inm  thick 
mounted  with  all  its  edges  in  a  clamped  arrangement  in  a  large  aluminum  base. 
The  aluminum  base  with  mounting  frame  sits  on  top  of  a  water  tank.  One  side  of 
the  base  plate  is  partially  treated  with  the  ACLD/AC/PCLD  and  the  other  side  is 
in  contact  with  water.  The  material  properties  and  thickness  of  piezo-electric 
material  and  the  viscoelastic  layer  listed  in  Table  (1).  The  size  of  the  combined 
piezo-electric  and  viscoelastic  patch  occupied  one-third  of  the  surface  area  of  the 
base  plate  and  it  is  placed  in  the  middle  of  plate  as  shown  in  Figure  (3-b).  A  laser 


1264 


sensor  is  used  to  measure  the  vibration  of  the  treated  plate  at  node  27  as  shown  in 
Figure  (3-b).  The  sensor  signal  is  sent  to  a  spectrum  analyzer  to  determine  the 
frequency  content  and  the  amplitude  of  vibration.  The  signal  is  also  sent  via 
analog  power  amplifiers  to  the  piezo-electric  layers  to  actively  control  the  sound 
radiation  and  structural  vibration.  The  radiated  sound  pressure  level  into  the  tank 
is  monitored  by  a  hydrophone  located  at  5.0  cm  below  the  plate  center.  This 
position  is  chosen  to  measure  the  plate  mode  (1,  1)  which  dominates  the  sound 
radiation.  The  hydrophone  signal  is  sent  also  to  the  spectrum  analyzer  to 
determine  its  frequency  content  and  the  associated  sound  pressure  levels. 


Figure  (3)  -  The  experimental  set-up,  (a)  schematic  drawing,  (b)  finite  element 
meshes. 


Table  (1)  -  Physical  and  geometrical  properties  of  the  ACLD  treatment 


Layer 

Thickness(m) 

Density  (Kg/m^) 

Modulus(MPa) 

viscoelastic 

S.OSxlO’" 

1104 

30** 

piezoelectric 

28xl0-‘ 

1780 

*  Young’s  modulus  **  Shear  modulus 


4.2  Experimental  Results 

Experimental  validation  of  the  dynamic  finite  element  model  of  the 
ACLD/plate  system  in  air  has  been  presented  by  Baz  and  Ro  [15]  in  detail.  Close 
agreement  was  obtained  between  the  theoretical  predictions  and  the  experimental 


1265 


measurements.  The  d)mamic  finite  element  model  is  therefore  valid  to  provide 
accurate  predictions. 

For  the  uncontrolled  treated  plate/fluid  system,  considered  in  this  study, 
the  experimental  results  indicate  that  coupling  the  plate  with  the  fluid  loading 
results  in  decreasing  the  first  mode  of  vibration  fi*om  59.475Hz  to  10.52Hz.  The 
coupled  finite  element  model  predicts  the  first  mode  of  vibration  to  decrease  firom 
57.91Hz  to  10.24Hz.  The  results  obtained  indicate  close  agreement  between  the 
theory  and  experiments. 

Figure  (4-a)  shows  a  plot  of  the  normalized  experimental  vibration 
amplitudes  for  the  fluid-loaded  plate  with  the  ACLD  treatment  using  different 
derivative  feedback  control  gains.  According  to  Figure  (4-a),  the  experimental 
results  obtained  by  using  the  ACLD  treatment  indicate  that  amplitude  attenuations 
of  1 1.36%,  48.25%  and  75.69%  are  obtained,  for  control  gains  of  2500,  5000,  and 
13500,  respectively.  The  reported  attenuations  are  normalized  with  respect  to  the 
amplitude  of  vibration  of  uncontrolled  plate,  i.e.  the  plate  with  PCLD  treatment. 
Figures  (4-b)  display  the  vibration  amplitudes  of  the  plate/fluid  system  with 
AC/PCLD  treatment  at  different  derivative  feedback  control  gains.  The 
corresponding  experimental  attenuations  of  the  vibration  amplitude  obtained  are 
4.6%,  20.29%,  54.04%  respectively. 


Figure  (4)  -  Effect  of  control  gain  on  normalized  amplitude  of  vibration  of  the 
treated  plate,  (a)  ACLD  control  and  (b)  AC/PCLD  control. 


Figures  (5-a)  and  (5-b)  show  the  associated  normalized  experimental 
sound  pressure  levels  (SPL)  using  ACLD  and  AC/PCLD  controllers,  respectively. 
The  normalized  experimental  SPL  attenuations  obtained  using  the  ACLD 
controller  are  26.29%,  50.8%  and  76.13%  compared  to  10.02%,  24.52%  and 
53.49%  with  the  AC/PCLD  controller  for  the  considered  control  gains.  Table  (2) 


1266 


lists  the  maximum  control  voltages  for  the  ACLD  and  AC/PCLD  controllers  for 
the  different  control  gains. 

It  is  clear  that  increasing  the  control  gain  has  resulted  in  improving  the 
attenuations  of  the  plate  vibration  and  the  sound  radiation  into  the  fluid  domain. 
It  is  evident  that  the  ACLD  treatment  has  produced  significant  vibration  and 
sound  pressure  level  attenuation  as  compared  to  the  attenuations  developed  by  the 
AC/PCLD  or  PCLD  treatments.  It  is  also  worth  emphasizing  that  the  ACLD 
treatment  requires  less  control  energy  than  the  conventional  AC/PCLD  treatments 
to  control  the  sound  radiation  from  the  plate. 


Figure  (5)  -  Effect  of  control  gain  on  normalized  sound  pressure  level  radiated 
from  the  treated  plate,  (a)  ACLD  control  and  (b)  AC/PCLD  control. 


Table  (2)  -  Maximum  control  voltage  for  the  ACLD/ AC/plate  system 


K. 

2500 

5000 

13500 

ACLD 

0 

21.75  V 

31.20V 

39.60V 

AC 

50.40V 

76.38V 

Figure  (6)  shows  the  mode  shapes  of  the  first  four  modes  of  the  treated 
plate  with  and  without  fluid-loading  as  obtained  experimentally  using 
STARMODAL  package.  Figure  (7)  shows  the  corresponding  theoretical 
predictions  of  the  first  four  mode  shapes.  Close  agreement  is  found  between 
experimental  measurement  and  theoretical  predictions. 

Figure  (8)  presents  comparisons  between  the  theoretical  and  experimental 
natural  frequencies  and  the  loss  factor  of  a  plate  treated  with  the  ACLD  and 
AC/PCLD  for  different  control  gains.  Close  agreement  between  theory  and 


1267 


experiment  is  evident.  Note  also  that  increasing  the  control  gain  has  resulted  in 
increasing  the  damping  ratio  for  both  ACLD  and  AC/PCLD  treatments.  The 
comparisons  emphasize  the  effectiveness  of  the  ACLD  treatment  in  acquiring  the 
large  damping  ratio  to  attenuate  the  structural  vibration  and  sound  radiation. 


Figure  (6)  -  Experimental  results  of  first  four  mode  shapes  of  treated  plate  (a) 
without  fluid  loading  and  (b)  with  fluid  loading. 


Figure  (7)  -  Theoretical  predictions  of  first  four  mode  shapes  of  treated  plate  (a) 
without  fluid  loading  and  (b)  with  fluid  loading. 


5.  SUMMARY 

This  paper  has  presented  theoretical  and  experimental  comparisons 
between  the  damping  characteristics  of  plates  treated  with  ACLD  and 


1268 


conventional  AC  with  PCLD  treatments.  The  dynamic  characteristics  of  the 
treated  plates  when  subjected  to  fluid  loading  is  determined  for  different 
derivative  control  gains.  The  fundamental  issues  governing  the  performance  of 
this  class  of  smart  structures  have  been  introduced  and  modeled  using  finite 
element  method.  The  accuracy  of  the  developed  finite  element  model  has  been 
validated  experimentally.  The  effectiveness  of  the  ACLD  treatment  in  attenuating 
structural  vibration  of  the  plates  as  well  as  the  sound  radiated  from  these  plates 
into  fluid  domain  has  also  been  clearly  demonstrated.  The  results  obtained 
indicate  that  the  ACLD  treatments  have  produced  significant  attenuation  of  the 
structural  vibration  and  sound  radiation  when  compared  to  PCLD  and  to  AC  with 
PCLD.  Such  favorable  characteristics  are  achieved  with  control  voltages  that  are 
much  lower  than  those  used  with  conventional  AC  systems.  The  developed 
theoretical  and  experimental  techniques  present  invaluable  tools  for  designing  and 
predicting  the  performance  of  the  plates  with  different  damping  treatments  and 
coupled  with  fluid  loading  that  can  be  used  in  many  engineering  applications. 


♦  PCLD  ■  ACLD,  K:d=l 3500  □  AC,  K:d=  13500 

•  ACLD,Kd=2500  O  AC,  Kd=2500 
A  ACLD,Kd=5000  A  AC,  Kd=5000 


Theoretical  Natural  Frequency  (Hz)  Theoretical  Damping  Ratio 

Figure  (8)  -  Comparison  between  theoretical  predictions  and  experimental  results, 
(a)  natural  frequency,  (b)  damping  ratio. 

ACKNOWLEDGMENTS 

This  work  is  funded  by  The  U.S.  Army  Research  Office  (Grant  number 
DAAH-04-93-G-0202).  Special  thanks  are  due  to  Dr.  Gary  Anderson,  the 
technical  monitor,  for  his  invaluable  technical  inputs. 


1269 


REFERENCES 

1.  Lindholm  U.  S.,  Kana,  D.  D.,  Chu,  W.  H.  and  Abramson,  H.  N.,  Elastic 
vibration  characteristics  of  cantilever  plates  in  water.  Journal  of  Ship 
Research,  1965, 9,  11-22. 

2.  Fu,  Y.  and  Price,  W.  G.,  Interactions  between  a  partially  or  totally  immersed 
vibrating  cantilever  plate  and  the  surrounding  fluid.  Journal  of  Sound  and 
Vibration,  1987, 118(3),  495-513. 

3.  Ettouney,  M.  M.,  Daddazio,  R.  P.  and  Dimaggio,  F.  L.,  Wet  modes  of 
submerged  structures  -  part  litheory.  Trans,  of  ASMS,  Journal  of  Vibration 
and  Acoustics,  1992, 114(4),  433-439. 

4.  Kwak,  M.  K.,  Hydroelastic  vibration  of  rectangular  plates.  Trans,  of  ASME 
Journal  of  Applied  mechanics,  1996,  63(1),  110-115. 

5.  Marcus,  M.  S.,  A  finite-element  method  applied  to  the  vibration  of 
submerged  plates.  Journal  of  Ship  Research,  1978,  22,  94-99. 

6.  Chowdury,  P.  C.,  Fluid  finite  elements  for  added  mass  calculations. 
International  Ship  Building  Progress,  1972, 19,  302-309. 

7.  Muthuveerappan  G.,  Ganesan,  N,  and  Veluswami,  M.  A.,  A  note  on  vibration 
of  a  cantilever  plate  immersed  in  water.  Journal  of  Sound  and  Vibration, 
1979,  63(3),  385-391. 

8.  Rao,  S.  N.  and  Ganesan,  N.,  Vibration  of  plates  immersed  in  hot  fluids. 
Computers  and  structures,  1985,  21(4),  111-1%! . 

9.  Everstine  G.  C.,  Prediction  of  low  frequency  vibrational  frequencies  of 
submerged  structures.  Trans,  of  ASME,  Journal  of  Vibration  and  Acoustics, 
1991, 113(2),  187-191. 

10.  Jones,  I.  W.  and  Salerno,  V.  L.,  The  vibration  of  an  internally  damped 
sandwich  plate  radiating  into  a  fluid  medium.  Trans,  of  ASME,  Journal  of 
Engineering  for  Industry,  1965,  379-384. 

11.  Sandman  B.  E.,  Motion  of  a  three-layered  elastic-viscoelastic  plate  under 
fluid  loading.  J.  of  Acoustical  Society  of  America,  1975,  57(5),  1097-1107. 

12.  Dubbelday,  P.  S.,  Constrained-layer  damping  analysis  for  flexural  waves  in 
infinite  fluid-loaded  plates.  Journal  of  Acoustical  Society  of  America,  1991, 
(3),  1475-1487. 

13.  Gu,  Y.  and  Fuller,  C.  R.,  Active  control  of  sound  radiation  from  a  fluid- 
loaded  rectangular  uniform  plate.  Journal  of  Acoustical  Society  of  America, 
1993,  93(1),  337-345. 

14.  Shen,  I.  Y.,  Bending  vibration  control  of  composite  plate  structures  through 
intelligent  constrained  layer  treatments.  Proc.  of  Smart  Structures  and 
Materials  Conference  on  Passive  Damping  ed.  C.  Johnson,  1994,  Vol.  2193, 
115-122,  Orlando,  FL. 


1270 


15.  Baz,  A.  and  Ro,  J.,  Vibration  control  of  plates  with  active  constrained  layer 
damping.  Journal  of  Smart  Materials  and  Structures,  1996,  5,  272-280. 

16.  Craggs,  A.,  The  transient  response  of  a  coupled  plate-acoustic  system  using 
plate  and  acoustic  finite  elements.  Journal  of  Sound  and  Vibration,  1971, 
15(4),  509-528. 

APPENDIX 

1.  Stiffness  Matrix  of  the  Treated  Plate  Element 

The  stiffoess  matrix  [Kp];  of  the  ith  element  of  the  plate/ACLD  system  is 
given  by  Baz  and  Ro  [15]: 

(A-1) 

where  [K,l  and  [KJi  denote  the  in-plane,  shear  and  bending  stifj&iesses  of 
the  ith  element.  These  stiffoess  matrices  are  given  by: 

j  =  layer  1,2, and 3  (A-2) 

and  [K.,1  =  I  jjB,]"[Dj,][B,]dxdy  j  =  layer  1, 2 and3  (A-4) 

with  G2  denoting  the  shear  modulus  of  the  viscoelastic  layer  and  the  matrices  [BJ, 


b1  =  :^ 


\({N2}-{N4)/d  +  {N,} 

(n,1  +fNj 


’  [Bb]  = 

2{N5},„_ 

{Naj 

[B.]  = 

{n. 

[®^p]  = 


(N.},.  +  {N3},.+h{Ns},„ 

{n.},,  +  +  {N2}^  +{n.)  „  +h{N4^, 


1271 


- 1 

0 

_ i 

1 - 

0 

Ei 

1  n 

and  fj,  1  EA 

1  n 

1-vJ 

Vj  1  u 

Vj  1  u 

I_ 

'  «  '“Vi 

0 

0 

i' 

_o  0  V. 

.  j=l,2and3  (A-5) 


where  h  =  (hi-h3)/2  and  d  =  (h2+hi/2+D)  with  D  denoting  the  distance  from  the 
mid-piane  of  the  plate  to  the  interface  with  the  viscoelastic  layer.  Also,  Ij 
represent  the  area  moment  of  inertia  of  the  jth  layer. 

2.  Mass  Matrix  of  the  Treated  Plate  Element 

The  mass  matrix  [Mp];  of  the  ith  element  of  the  plate/ACLD  system  is 
given  by: 

(A-6) 

where  [Mjp]j  and  [M^Jj  denote  the  mass  matrices  due  to  extension  and  bending  of 
the  ith  element.  These  matrices  are  given  by 

[Mi,].=p,h,££({N,}^{N,}+{N3}''{N,})dxdy  +  p,h,[  [({N3)'"{N3}  +  {NX{N4))dxdy 

+  ip2h2  £  {  +{N,}^{N,})dxdy 

and  [  ],  =  (p,h,  +  P3h3  +  P3h3)  £  £  [  N3  f  [  N3  ]dxdy  (A-7) 

where  {NJ  =  {N,}+{N3}+h{Ns},3  and  {N,}  =  {NJ+{N4}+h{N5},3 

3.  Control  Forces  and  Moments  Generated  by  the  Piezo-electric  Layer 

3.1  The  in-plane  piezo-electric  forces 

The  work  done  by  the  in-plane  piezo-electric  forces  {Fp}i  of  the  ith 
element  is  given  by: 

i{5'}-{Fp}rhi££%d>'dy  (A-8) 


where  j=l  for  ACLD  control  or  j=3  for  AC  control.  Also,  Ojp  and  Sjp  are  the  in¬ 
plane  stresses  and  strains  induced  in  the  piezo-electric  layers.  Equation  (A-8) 
reduces  to: 


1272 


='"•1 IKFK] 


for  k=l, 4 


3.2  The  piezo-electxic  moments 

The  work  done  by  the  piezo-electric  moments  {Mp}i  due  to  the  bending  of 
the  piezo-electric  layer  of  the  ith  element  is  given  by: 


5'  .MM 


=  hj  U,e,Ady 


(A-10) 


Where  Gjb  and  are  the  bending  stresses  and  strains  induced  in  the  piezo-electric 
layers.  Equation  (A-10)  reduces  to: 


Mpxk 

x.' 

Mpyk 

d32 

Mp,yk 

_  0 

fork=l,..,4 


(A-11) 


4.  Stiffness  and  Mass  Matrices  of  the  Fluid  Element 

The  stiffness  matrix  [KJj  and  mass  matrix  [MJ^  of  the  ith  element  of  the 
fluid  system  are  given  by: 


[B,rNdv 


(A-12) 

(A-13) 


[N,]>,]dv  (A-13) 

where  [b^ ]  =  [[N^],,  [N^]  ^  [N^]  and  c  is  the  sound  speed. 

5.  Coupling  matrix  of  the  Treated  Plate/Fluid  System 

The  coupling  matrix  [Q]  of  the  interface  element  of  the  structure/fluid 
system  can  be  presented  by: 

[q]=  f  [  {N5}''[Nf]dxdy  (A-14) 

Ja  Jb 


1273 


1274 


ANALYTICAL  METHODS  II 


DYNAMIC  RESPONSE  OF  SINGLE-LINK  FLEXIBLE 
MANIPULATORS 


E.  Manoach^  G.  de  Paz^  K.  Kostadinov^  and  F.  Montoya^ 

^  Bulgarian  Academy  of  Sciences,  Institute  of  Mechanics 
Acad.  G.  Bonchev  St.  Bl.  4;  1113  Sofia,  Bulgaria 
^  Universidad  de  Valladolid,  E.T.S.I.L  Dpto.  IMEIM. 

C/Paseo  del  Cauce,  s/n  47011-  Valladolid,  Spain 

1.  INTRODUCTION 

The  flexible-link  manipulators  have  many  advantages  over  the  traditional  stiff 
ones.  The  requirements  for  light-weight  and  energy  efficient  robotic  arms 
could  be  naturally  satisfied  by  using  flexible  manipulators.  On  the  other  hand 
the  application  of  the  robotic  arm  in  such  activities  as  positioning  in  electronic 
microscopes  and  disc-drivers,  hammering  a  nail  into  a  board  or  playing  tennis, 
also  forces  the  modeling  and  control  of  the  dynamic  behavior  of  flexible  link 
manipulators. 

In  most  cases  the  elastic  vibrations  which  arise  during  the  motion  must  be 
avoided  when  positioning  the  end  point  of  a  robotic  arm.  These  are  a  part  of 
the  reasons  that  cause  a  great  increase  of  the  publications  in  this  topic  in  recent 
years. 

In  most  papers  the  flexible  robotic  arms  are  modeled  as  thin  linear  elastic 
beams.  In  [1-3]  (and  many  others)  the  Bemouli-Euiler  beam  theory,  combined 
with  finite-element  technique  for  discretization  with  respect  to  the  space 
variables  is  used  for  modeling  and  control  of  single-link  flexible  manipulators. 
The  same  beam  theory,  combined  with  mode  superposition  technique  is  used 
in  [4].  Geometrically  nonlinear  beam  theories  are  used  in  [5,6]  for  the 
modeling  of  a  single-link  and  multi-link  flexible  robotic  manipulators, 
correspondingly. 

Taking  into  account  the  fact  that  robotic  arms  are  usually  not  very  thin  and  that 
the  transverse  shear  could  play  an  important  role  for  dynamically  loaded 
structures  [7]  the  application  of  the  Bernouli-Euiler  beam  theory  could  lead  to 
a  discrepancy  between  the  robotic  arm  behavior  and  that  one  described  by  the 
model. 


1275 


The  aim  of  this  work  is  to  model  the  dynamic  behavior  of  a  single  link  flexible 
robotic  arm  employing  the  Timoshenko  beam  theory,  which  considers  the 
transverse  shear  and  rotary  inertia.  The  arm  is  subjected  to  a  dynamic  loading. 
As  in  [3],  the  viscous  friction  is  included  into  the  model  and  slip-stick 
boundary  conditions  of  the  rotating  hub  are  introduced.  Besides  that,  the 
possibility  of  the  rise  of  a  contact  interaction  between  the  robotic  arm  and  the 
stop  (limiting  support)  is  included  into  the  model.  The  beam  stress  state  is 
checked  for  plastic  yielding  during  the  whole  process  of  deformation  and  the 
plastic  strains  (if  they  arise)  are  taken  into  account  in  the  model.  The 
numerical  results  are  provided  in  order  to  clarify  the  influence  of  the  different 
parameters  of  the  model  on  the  response  of  the  robotic  arm. 

2.  BASIC  EQUATIONS 
2.1.  Formulation  of  the  problem 

The  robotic  arm  -  flexible  beam  is  attached  to  a  rotor  that  has  friction  and 
inertia.  The  beam  is  considered  to  be  clamped  to  a  rotating  hub  and  its  motion 
consists  of  two  components:  “rigid-body”  component  and  a  component 
describing  the  elastic  deflection  of  the  beam  (see  Figure  1).  The  motion  of  the 
flexible  beam  is  accomplished  in  the  horizontal  plane  and  gravity  is  assumed 
to  be  negligible. 


Figure  1.  Model  of  one-link  flexible  manipulator. 


1276 


Tip  of  the  beam  (with  attached  tip-mass)  is  subjected  to  an  impulse  loading. 
Stick-slip  boundary  conditions  due  to  Coulomb  friction  of  the  hub  are 
introduced  when  describing  the  motion  of  the  beam.  In  other  words,  if  the 
bending  moment,  about  the  hub  axis,  due  to  the  impact  is  lower  than  the  static 
friction  torque  threshold  then  the  hub  is  considered  clamped  and  the  beam 
elastic  motion  is  considered  only.  When  the  bending  moment  exceeds  friction 
torque  threshold  this  boundary  condition  is  removed,  allowing  rotation  of  the 
hub  and  the  arm.  When  the  hub  speed  and  kinetic  energy  of  the  beam  become 
again  beneath  the  torque  threshold,  the  hub  clamps  again. 

The  possibility  of  the  rise  of  a  contact  interaction  between  the  robotic  and  the 
stop  is  envisaged.  In  this  case,  if  the  hub  angle  exceeds  the  limit  value,  the 
robotic  arm  clamps  and  a  part  of  the  beam  goes  in  a  contact  with  the  stop, 
which  is  modeled  as  an  elastic  foundation  of  a  Vinkler  type. 

In  view  of  the  fact  that  the  impact  loading  and  contact  interactions  are 
included  in  the  investigations,  it  is  expedient  to  be  considered  the  rise  of 
plastic  strains  in  the  beam. 


2.2.  Deriving  the  equations  of  motion. 


The  total  kinetic  and  potential  energy  of  the  rotating  hub  with  the  attached 
beam  (described  by  the  Timoshenko  beam  theory)  and  a  lumped  mass  at  it’s 
tip  can  be  expressed  as  follow: 

Ek  =^|pA[u(x,t)]  dx  +  |j„[e(t)]^+tMT[u(l,t)f jEl[(()(x,t)]  dx 


EIl^l  +kGA’ 
dx. 


V  dx 


dx 


(1  a,b) 


In  these  equation  u(x,t)  is  the  total  displacement 
u(x,t)  =  w(x,t)+x0(t)  (2) 

and  w  is  the  transverse  displacement  of  the  beam,  (p  is  the  angular  rotation, 
9  is  the  hub  angle,  E  is  the  Young  modulus,  G  is  the  shear  modulus.  A:  is  a 
shear  correction  factor,  p  is  the  material  density,  Jh  is  the  inertia  moment  of 
the  hub,  A=b*h  is  the  beam  cross-section  area,  h  is  the  thickness,  b  is  the 
width,  I=bh^/12 , 1  is  the  length  of  the  beam,  Mt  is  the  tip  mass,  t  is  the  time. 


1277 


Denoting  the  work  of  external  forces  (applied  actuating  torque  T(t)  and  the 
beam  loading  p(x,t))by 


W=T(t)e(t)+  \p{x,t)w{,x,t)dx 

0 

the  Hamilton's  principle  can  be  applied: 
j5(E,-Ep)dt+  j5Wdt  =  0 


(3) 

(4) 


Substituting  eqns  (l)-(3)  into  eqn  (4)  after  integrating  and  including  damping 
of  the  beam  material,  the  viscous  friction  of  the  hub  and  the  reaction  force  of 
the  elastic  foundation  the  following  equations  of  motion  can  be  obtained: 


.  r  .  X.  i2x  de  ,  ,5^w(l,t)  , 

(J  +  Jh  +  M-j-1  ^  j  ^2  ^  ^  ^  ^2 


i 

JpAx 


a^w(x,t) 

dt^ 


dx  =  T(t) 


EI^-^  +  kGA 
dx^ 


dw 


dx 


dw 


.d^cp 


- 9  “  <^2 - — 2  ~  0<x</ ,  t>0  (5a-c) 


dt 


dt 


kGA 


+  x- 


a^0 

'St" 


-R(x,t)  =  p(x,t) 


The  boundary  conditions  are: 
w(0,t)=(p(0,t)=0 


dcpgt)  Q 
dx 


kGA 


dw(l,t) 

dx 


d^w(lt) 

dt^ 


(6a-c) 


and  the  initial  conditions  are: 

wfx,  0)  =w^(x),  w(x,0)  =  w°  (x) ,  p(x, 0)=(f^(x),  (p(x,0)  =  9°  (x) 

e(t)  =  e(t)  =  o,  t<t,,p 

or  9(0  = 

when  |9(t)|  and  kinetic  energy  of  the  beam  falls  simultaneously  under  the 
Coulomb  friction  thresholds. 

In  eqns  (5)  the  viscous  friction  coefficient  of  the  hub  is  denoted  by  di,  d2  and 
d^  are  damping  coefficients  of  the  beam  material,  J  is  the  moment  of  inertia  of 
the  beam  about  the  motor  axis  and  R(x,t)  is  the  reaction  force  of  the  stop 
disposed  from  Xj  to  X2  (xj<X2<l)  and  modeled  as  an  elastic  foundation  with 
Vinkier  constant  r: 


(7a-d) 
(7  e,f) 
(7e)’ 


1278 


R(x,t)  = 


r[w(x,  t)  -  (x)]  for  x  j  <  x  <  x^ ;  |0l  > ' 

0  for  0<x<x,,X2  <x</;  iBi  < 


The  time  when  the  bending  moment  about  the  hub  axis  exceeds  the  friction 
torque  threshold  is  denoted  by  tsUp^  When  t>tsiip  this  condition  is  removed 
(allowing  rotation  of  the  hub  and  the  beam)  until  the  moment  when  the  beam 
clamps  again. 

2.3.  Elastic-plastic  relationships 

The  beam  stress-strain  state  is  usually  expressed  in  terms  of  generalized 
stresses  and  strains  which  are  function  of  x  coordinate  only.  As  a  unique 
yield  criterion  in  terms  of  moments  and  the  transverse  shear  force  does  not 
exist  according  to  Drucker  [8],  the  beam  cross-section  is  divided  into  layers 
and  for  each  of  them  the  stress  state  has  to  be  checked  for  yielding. 

The  relation  between  the  stress  vector  S  =  and  the  strain  vector 

s  =  |-z  ^ I  ’  generally  presented  as 


S=[D]8, 

where  in  the  case  of  an  elastic  material  [D]  =  [D^]  = 


is  the  elastic 


matrix  and  f(z)  is  a  function  describing  the  distribution  of  the  shear  strains 
along  the  thickness. 

On  the  basis  of  the  von  Mises  yield  criterion,  the  yield  surface  is  expressed 


After  yielding  during  infinitesimal  increment  of  the  stresses,  the  changes  of 
strains  are  assumed  to  be  divisible  into  elastic  and  plastic  parts 


As  =  As^  +  As^ 


where 


As 


1279 


By  using  eqns  (10),  (11)  and  the  associated  flow  rule  [9],  following  Yamada 
and  others  [10],  the  following  explicit  relationship  between  the  increments  of 
stresses  and  strains  is  obtained 


.  AS  =  [d‘'’]A8 

where  [-0'^]  is  the  elastic-plastic  matrix: 


HY\d^] 

LjaLiVi 

r^r 

1  J  1  J  L  j\asj 

1 - 

C/2 

1— 

108] 

In  this  equation  H  is  a  function  of  the  hardening  parameter.  For  ideal  plasticity 
H  is  equal  to  zero,  while  for  a  wholly  elastic  material  H  ->  co  . 

3.  Use  of  mode  superposition. 

3.1.  Rearrangement  of  the  equation  of  motion. 

Let  the  total  time  interval  T  on  which  the  dynamic  behavior  of  the  structures  is 
investigated,  be  divided  into  sequence  of  time  increments  y+J. 

In  the  numerical  calculations  the  following  dimensionless  variables  are  used: 
x  =  xl  U  w  =  w//,  t  =tl  / c  c  =  /  p 

and  then  omitting  the  bars,  and  after  some  algebra,  the  governing  equations 
can  be  written  in  the  following  form: 


d'e  de 

- T-  +  C,  - 

d  t^  '  d  t 


--cJ  w(l,t)-  f(p(x,t)dx  =C3  T(t)+  fxp(x,t)dx 
tv  0  /Vo  / 


09  0^9  -I  c/w  I 


(15  a-c) 


0^W  01V  0^iv  09 

dt^  ^  dt  1 0x  ^  dx 


=  -p~G{-G^ 


where  a=12P/h^  ,  p~kG/E,  p=pI/(EA),  c,=  d^l/(cJ}{),  Arp/(JHp)  C3= 
P/(c^J}^  ),  C4~d49/EI,  C3=djl/EA,  =5/6.  The  nonlinear  force  due  to  the 

reaction  of  the  foundation  is  denoted  by  G^  =  R(x,t).l/EA  and  Gf  and  G2  are 
the  components  of  the  so-called  non-linear  force  vector  Gp(  .G^}  which  is 
due  to  the  inelastic  strains.  It  has  the  presentation  (see  [7] ): 


1280 


3.2  Mode  superposition  method 

The  l.h.s.  of  eqns  (15  b,c)  is  a  linear  form  and  therefore  the  mode 
superposition  method  can  be  used  for  its  solution.  As  the  eigen  frequencies 
and  the  normal  modes  of  vibrations  of  an  elastic  beam  do  not  correspond  to  the 
real  nonlinear  system,  these  modes  are  called  "pseudo-normal"  modes. 

Thus,  the  generalized  displacements  vector  v  =  |a”^  (p,  w|  is  expanded  as  a 
sum  of  the  product  of  the  vectors  of  pseudo-normal  modes  v^  and  the  time 
dependent  functions  q„(t)  as 

(17) 

n 

The  analytically  obtained  eigen  functions  of  the  elastic  Timoshenko  beam  are 
chosen  as  basis  functions  (see  APPENDIX).  When  the  tip  mass  is  attached  to 
the  beam  the  eigen  functions  of  the  system  used  in  the  mode  superposition 
method  are  preliminary  orthogonalized  by  standard  orthogonalization 
procedure  as  it  is  mentioned  in  the  APPENDIX. 

Substituting  eqn  (17)  into  (15  b,c),  multiplying  by  v^(x),  integrating  the 
product  over  the  beam  length,  invoking  the  orthogonality  condition  (see  (A9) ) 
and  assuming  a  proportional  damping  for  the  beam  material 

|(c4(pf,  +c^wl)rdr  the  following  system  of  ordinary  differential 

equations  (ODE)  for  0(t)  and  qn(t)  is  obtained: 


^(f)  +  c,  (9(0  =  Cj  2]  [«'» (1)  -  ]?» 


^  » 


+  C3(r(0  +  P(0) 


(18a, b) 


9„(o+2f„ffl„9„(o+®,k,(o=-‘y„  -'f„&(o+&!’ 

In  this  equations 

1 

11  V  f 

Y„  =  J(p„  6„  =  =  jxw,Xx)ck,  P(t)=  \xp{x,t)  dx , 


1  1 

g;(t)=  |G'’(x,t)v„(x)dx,  g',(t)=  jG'(x.t)w„(x)dx,  co„  are  the  eigen 

0  0 

frequencies  of  the  elastic  clamped  beam  and  are  the  modal  damping 
parameters. 

The  initial  conditions  defined  by  eqns  (7  a-d)  are  transformed  also  in  terms  of 
q„(0),  and  ?„(0) 

9„(0)  =  9„“,  4„(0)  =  4°, 

q°„  =  J(w°w„  t-a'VVjdx,  (j“„  =  j(w"w„ +a-'(p°(p„)dx,  (19a-d) 

0  0 

The  obtained  system  of  nonlinear  ordinary  differential  equations  is  a  stiff  one 
and  it  is  solved  numerically  by  the  backward  differential  formula  method,  also 
called  the  Gear's  method  [11]. 

The  rise  of  plastic  strains  is  taking  into  account  by  using  an  iterative  procedure 
based  on  the  "initial  stresses"  numerical  approach  [7]. 


4.  RESULTS  AND  DISCUSSION 

Numerical  results  were  performed  for  the  robotic-arm  with  the  same  material 
and  geometrical  characteristic  as  these  given  in  [3]  in  order  to  make  some 
comparisons.  Model  parameters  are:  E=6.5xl0  Pa  ,v=0.2,  Cp=2.6xl0  Pa, 
1=0.7652,  b=0.00642  m,  h=0.016  m,  p=2590  kgW  ,  Mt=0,153  kg,  Jh=0.285 
kgm^  (Jh  is  not  defined  in  [3].) 

The  aim  of  the  computations  is  to  show  and  clarify  the  influence  of  the  elastic 
or  elastic-plastic  deformation  on  the  motion  of  the  robotic-arm,  to  demonstrate 
the  effect  of  the  hub  friction,  slip-stick  boundary  conditions  and  the  contact 
interaction  between  the  beam  and  the  stop. 


1282 


Only  impact  loading  on  the  beam  is  considered  in  this  work,  i.e.  dynamic  load 
p(x,t)  and  applied  torque  T(t)  are  equal  to  zero.  The  impact  loading  is 
expressed  as  an  initial  velocity  applied  to  the  tip  of  the  beam  0.95  <  x  <  1 . 

Nine  modes  are  used  in  expansion  (15)  but  the  results  obtained  with  number  of 
modes  greater  than  nine  are  practically  indistinguishable  from  these  shown 
here. 

For  all  calculations  the  material  damping  is  equal  to  8%  of  the  critical 
damping. 

The  results  for  the  rotation  of  the  hub  of  the  flexible  manipulator  with  an 
attached  mass  at  its  tip  -1  and  without  an  attached  mass  -2  are  shown  in  Figure 
2.  The  hub  friction  is  not  considered.  The  beam  is  subjected  to  an  impact 
loading  with  initial  velocity  =-1.95  m/s.  As  can  be  expected,  the  hub  angle 
increases  much  faster  in  the  case  of  the  beam  with  an  attached  mass.  The 
corresponding  beam  deflections  are  presented  in  Figure  3.  The  results  obtained 
are  very  close  to  these  obtained  in  [3]  (Fig.  7  and  8  in  [3]).  The  frequencies  of 
forced  vibrations  obtained  in  [3],  however,  correspond  to  the  beam  without  an 
attached  mass. 


0,  rad 


-4.00  I - ^ ^ ^ ^ - 1 - ^ ^ i ^ 

0,00  l.OO  2.00  3.00  4.00  5.00 

t,  sec 

Figure  2.  Hub  response  without 
viscous  friction.  1  -  beam  with  an 
attached  mass;  2-  without  an  attached 
mass 


w,  m 


Figure  3.  Deflection  of  the  tip  of  the 
beam  without  viscous  friction  of  the 
hub.  1  -  beam  with  an  attached  mass; 
2  -  without  an  attached  mass 


1283 


The  influence  of  the  hub  friction  on  the  flexible  manipulator  response  can  be 
seen  in  Figure  4.  The  viscous  friction  is  set  di=0.1  Nms,  the  static  Coulomb 
friction  threshold  is  equal  to  0.06  Nm  and  three  cases  of  the  hub  slip-stick 

threshold  are  tested:  |e|  stick  =  0-005  rad.s'^  -  1,  |0|  stick  =  0.0085  rad.s'^  , 

stick  =  0.01  rad.s'^  For  this  initial  velocity  (w°  =-1.95m/s)  the  hub  slips 

very  fast  from  the  initial  clamped  state  and  the  beam  begins  to  rotate.  As  can 
be  expected,  the  consideration  of  the  viscous  friction  of  the  hub  leads  to  a 
decrease  of  the  angle  of  rotation  of  the  beam  and  changes  the  linear  variation 
of  0  with  time.  The  results  show  also  that  the  value  of  the  hub  speed 

threshold  |0|  stick  exercises  an  essential  influence  on  the  motion  of  the 
rotating  system. 

When  |e|  =0.005  rad.s'  the 

beam  sticks  at  t  =  2.602  s  after  that 
the  hub  periodically  slips  and  sticks 
which  also  leads  to  damping  of  the 

motion.  When  joj  stick  “  0.0085 

rad.s'^  the  start  of  sticking  occurs  at 
t  =  1.7207  s  and  after  t=2.417  s  the 
hub  clamps  with  short  interruptions 
till  t=3.4  s  when  due  to  the  elastic 
vibrations  it  snaps  in  the  direction 
opposite  to  w°»  clamps  again  at 
3.679s,  slip  at  4.5s,  and  finally 

clamps  at  t=4.5s.  When  joj  stick 
=0.0085  rad.s"'  the  sticking  begins 
at  t= 1.0525s  and  very  fast  (at  t«2  s) 
the  beam  clamps  with  0=-O.587  rad. 

In  order  to  observe  the  occurrence  of  the  plastic  deformation  the  beam  was 
subjected  to  impulse  loading  having  larger  values  of  initial  velocities.  In 
addition,  the  contact  interaction  between  the  beam  and  the  stop  was 
considered. 

The  beam-tip  deflection  in  the  presence  of  a  contact  with  the  stop  disposed  at 
X  €[0.16,  0.263]  and  initial  velocity  w°  =-15.95  m/s  is  shown  in  Fig.  5. 


0,rad 


di=0.1Nms.  1-  |0|  stick”0-005rad.s"', 

2  -  |0|  stick  =  0-0085  rad  s"'  ; 

3-  |0|  stick  =0-0 is-' 


1284 


In  order  to  reduce  the  computational  time  the  limit  value  of  0  was  chosen 
e^j^p=0.0025  rad.  When  this  value  was  reached  the  problem  was  automatically 
resolved  with  new  initial  conditions  0=0,  w°=w(Xstop,tstop),  etc. 

As  can  be  seen,  the  presence  of  the  contact  interaction  during  the  process  of 
motion  of  the  beam  due  to  the  elastic  support  for  x  e[0.16,  0.263]  leads  to  a 
decrease  of  the  amplitudes  of  vibrations  in  the  direction  of  the  stop  in 
comparison  with  the  amplitudes  in  the  opposite  direction.  The  variation  of  the 
beam  displacements  along  the  beam  length  for  the  first  0.8  s  of  motion  is 
shown  in  Fig.  6.  It  must  be  noted  that  in  this  case  of  loading  a  plastic  yielding 
occurs.  It  is  assumed  that  beam  material  is  characterized  by  an  isotropic  linear 
strain  hardening  and  H=0.5.  The  plastic  yielding  occurs  simultaneously  with 
the  contact  interaction  at  t=0.01366  s  at  the  clamped  end  of  the  beam.  At 
t=0.0186  s  the  plastic  zone  spreads  to  x  =  0.158  and  at  t=0.08767  if  covers  the 
length  to  X  =0.3 1 .  The  last  points  that  yields  are  x  =0.55,  0.61  at  t=l .  1 19  s. 
Seven  layers  along  the  beam 
thickness,  symmetrically  disposed 
about  the  beam  axis  was  checked  for 
yielding  (N2=7)  but  the  plastic  zone 
has  reached  the  second  and  6th 
layers  only  at  the  clamped  end  of  the 
beam  (x  =0).  In  all  other  point  along 
beam  length  the  plastic  yielding 
occurs  only  at  the  upper  and  lower 
surface  of  the  beam.  The  plastic 
strains  are  small  and  the  response  of 
the  beam  is  not  very  different  from 
the  wholly  elastic  response. 

Nevertheless,  the  appearance  of  such 
kind  of  plastic  deformations  in  the 
structures  used  for  the  precise 
operations  must  be  taken  into 
account  in  the  manipulator  self 
calibration  procedure. 


w,m 

0.40 

0.30 

0.20 

0,10 

0.00 

-O.IO 

-0.20 


-0.30 


0.00 


1.00 


2.00 


3.00 


4.00  5.00 

t,  sec. 

Figure  5.  Deflection  of  the  tip  of  the 
beam  with  time  in  the  case  of  a 
contact  with  the  stop.  r=5.5xl0^  Pa 


5.  CONCLUSIONS 

In  this  work  a  model  describing  dynamic  behavior  of  a  deformable  beam 
attached  to  a  rotating  hub  that  has  friction  and  inertia  is  developed.  The 
Timoshenko  beam  theory  is  used  to  model  the  elastic  deformation  of  the  beam. 


1285 


Figure  6,  Variation  of  the  elastic-plastic  beam  displacement  along  the  beam 
length  with  time  in  the  case  of  a  contact  with  the  stop. 

The  slip-stick  boundary  conditions  are  also  incorporated  into  the  model 
The  possibility  of  the  rise  of  undesired  plastic  deformations  in  the  case  of  a 
high  velocity  impact  on  the  clamped  robotic  arm,  or  in  the  case  of  a  contact 
with  limiting  support  (stop)  is  included  into  the  model. 

The  analytically  obtained  eigen  functions  of  the  elastic  Timoshenko  beam 
vibrations  are  used  to  transform  the  partial  differential  equations  into  a  set  of 
ODE  by  using  the  mode  superposition  method.  This  approach  minimizes  the 
number  of  ODE  which  have  to  be  solved  in  comparison  with  another 
numerical  discretization  techniques  (finite  elements  or  finite  difference 
methods). 

The  results  obtained  show  the  essential  influence  of  elasticity  on  the  robotic- 
arm  motion. 

The  model  will  be  used  to  synthesise  a  control  of  one  link  flexible 
manipulators  and  for  a  self  calibration  procedure  when  plastic  deformation 
would  occur. 

Acknowledgments 

The  authors  gratefully  acknowledge  the  financial  support  from  EC  Copernicus 
Program  under  the  Project  ROQUAL  CIPA  CT  94  0109. 

The  first  author  wishes  to  thank  the  National  Research  Fund  for  the  partial 
financial  support  on  this  study  through  Contract  MM-5 17/95. 


1286 


REFERENCES 


1.  Bayo,  E.  A  finite-element  approach  to  control  the  end-point  motion  of  a 

single  -link  flexible  Robert.  J.  Robotic  System,  1987,  4„  63-75 

2.  Bayo,  E.  and  Moulin,  H.,  An  efficient  computation  of  the  inverse  dynamics 

of  flexible  manipulators  in  the  time  domain.  IEEE  Proc  Int.  Conf.  on 
Robotics  and  Automations,  1989,  710-15. 

3.  Chapnik,  B.V.,  Heppler,  G.R.,  and  Aplevich,  J.D.  Modeling  impact  on  a 

one-link  flexible  robotic  arm.  IEEE  Transaction  on  Robotics  and 
Automation,  1991,7,479-88. 

4.  Liu,  L.  and  Hac,  A.,  Optimal  control  of  a  single  link  flexible  manipulator. 

Advances  in  Robotics,  Mechatronics,  and  Haptic  Interfaces,  1993,  DCS- 
49,  303-13. 

5.  Wen,  J.T.,  Repko,  M.  and  Buche,  R.,  Modeling  and  control  of  a  rotating 

flexible  beam  on  a  translatable  base.  Dynamics  of  Flexible  Multibody 
Systems:  Theory  and  Experiment,  1992,  DCS-37,  39-45. 

6.  Sharan,  A.M.  and  Karla,  P.,  Dynamic  Response  of  robotic  manipulators 
using  modal  analysis.  Meek  Mach.  Theory,  1994,  29,  1233-49. 

7.  Manoach,  E.  and  Karagiozova,  D.  Dynamic  response  of  thick  elastic-plastic 

beams.  International  Journal  of  Mechanical  Sciences,  1993,  35,  909-19 

8.  Drucker,  D.C.  Effect  of  shear  on  plastic  bending  of  beams.  J.  of  Applied 

Mechanics,  1956, 23,  515-21 

9.  Hill,  R.  Mathematical  Theory  of  Plasticity,  1950,  Oxford  University  Press, 
London. 

10.  Yamada,  Y.,  Yoshimura,  N.  and  Sakurai  T.,  Plastic  stress-strain  matrix  and 
its  application  for  the  solution  of  elastic-plastic  problems  by  the  finite 
elements,  Int.  J.  of  Mechanical  Sciences  ,  1968, 10,  343-54 

11.  Gear,  C.W.,  Numerical  initial  value  problem  in  ordinary  differential 
equations,  1971,  Prentice-Hall,  Englewood  Cliffs,  NJ. 

12.  Abramovich,  H.  Elishakoff,  1.  Influence  of  shear  deformation  and  rotary 
inertia  on  vibration  frequencies  via  Love’s  equations.  J.  Sound  Vibr.,  1990, 
137,  516-22. 


1287 


APPENDIX:  NORMAL  MODES  OF  FREE  VIBRATIONS  OF  A 
CLAMPED  TIMOSHENKO  BEAM  WITH  AN  ATTACHED  MASS, 


Equations  (5  b,c)  can  be  decoupled,  transforming  them  into  two  fourth  order 
equations  [12]  as  regards  cp  and  w. 

Solving  this  equations  (with  p=0  and  R=0)  and  using  the  boundary  conditions 
(6)  (with  0=0) ,  the  equations  of  the  frequencies  and  forms  of  vibrations  of  the 
beam  are  obtained. 

Introducing  following  denotations 

1/2  (A1  a-e) 

^2„={®,^(l  +  P)  +  [®:a  +  P)^+4(B,^(a-pco,^,)]''^}  /2, 

/l„  =(4+CoJP)/5,„,  /2„  /3„  =(4-®SP)''-S!». 

the  frequencies  of  free  vibrations  are  determined  as  roots  of  equation: 


a)  In  the  case  >0  i.e.  co  <  a  /  p  the  frequencies  equation  is: 

^11^22  ”^12^21  “  ^  >  (-^2) 
where 

^11  ~ /2n‘^2n  ^12  ~  ■^/2«‘^2«  ®^^(‘^2«) 

h\  =  -f\nM.Sxn )  “  /2„  sin(52„ )  +  s^„sh(si, )  +  52„  sin(52„ )  +  ,  .  . 

(A3  a-d) 

M‘^lnCh(^lJ  +  ‘^2«  cos(,y2J] 


*22  =  /2j=OS(^2«)-cll('SlJ]  +  4^S,„ch(i,„)-52„  COS(52„)  + 

J\n 

+  sl„  sin(i2„)] 

Jin 


and  the  modes  of  vibrations  are: 

w„W  =  -S„ 

1 

'  b 


^(ch(i,„x)  -  cos(j2„x))  +  sin(i2„x)  -  ^sh(i|„x) 

J\» 


L^i 


(A4  a,b) 

(/,„sh(,s,„x)  +  sin(,y2«^))  +  fi,,  icos{s^„x)  -  ch(s,„x) 


b)  In  the  case  5^,,  <0  i.e.  >  a  /  P  the  eigen  frequency  equation  (A2)  has 
the  following  presentation: 


1288 


b\\  =  +  /2„i2„  C0S(J2„) 

bn  =  ?i„/2nSin(?i„)  -  /2„i2n  sin(j2„) 

621  =  /3„sin(ii„)  -  /2„  sin(52„)  -?i„sin(Ji„)  +  S2„  sinC^jJ  - 
>.[J,^„COS(  j,„  )  -  sl„  C0S(i2„)] 

*22  =  /2„[cOS(S2„)  -  COS(J,„)]  -  ^  S,„COS(?,„)  +  S2„  COS{s-^„)  - 

J\n 

-  sl„  sin(52„)] 

7  In 

sh  =  -5?„ 


and  the  of  vibrations  are: 

w„(x)  = 


f  b 

sin(j2„2c)  -  ^sin(?,„x)  -  ^(cosCij,,*)  -  cos(j,„x)) 
/i„  *11 


'P„W  =  -S» 


(A5  a-e) 


(A6a-b) 


/j,,  (cos(i2„x)  -  cos(^|„x))  -  ^(/3„sin(i,„x)  -  f^„  sin(jj„x)) 

0,, 


When  a  mass  is  not  attached  at  the  beam  tip  the  following  orthogonality 
condition  is  fulfilled: 

1 


p  fO,  n^  m; 

(a  _ 

J  [1,  n  -  m. 


(A7) 


and  when  an  attached  mass  is  considered  the  modes  are  orthogonalized  by 
standard  orthogonalization  procedure. 

The  constants  are  obtained  from  condition  (A7). 


1289 


1290 


Wave  Reflection  and  Transmission 
in  an  Axially  Strained,  Rotating  Timoshenko  Shaft 


B.  Kang  ‘  and  C.  A.  Tan  ^ 
Department  of  Mechanical  Engineering 
Wayne  State  University 
Detroit,  Michigan  48202,  U.  S.  A. 


Abstract 

In  this  paper,  the  wave  reflection  and  transmission  characteristics  of  an  axially  strained, 
rotating  Timoshenko  shaft  under  general  support  and  boundary  conditions,  and  with  geometric 
discontinuities  are  examined.  The  static  axial  deformation  due  to  an  axial  force  is  also  included 
in  the  model.  The  reflection  and  transmission  matrices  for  incident  waves  upon  these  point 
supports  and  discontinuities  are  derived.  These  matrices  are  combined,  with  the  aid  of  the 
transfer  matrix  method,  to  provide  a  concise  and  systematic  approach  for  the  free  vibration 
analysis  of  multi-span  rotating  shafts  with  general  boundary  conditions.  Results  on  the  wave 
reflection  and  transmission  coefficients  are  presented  for  both  the  Timoshenko  and  the  simple 
Euler-Bemoulli  models  to  investigate  the  effects  of  the  axial  strain,  shaft  rotation  speed,  shear 
and  rotary  inertia. 


‘  Graduate  Research  Assistant.  Tel:  +1-313-577-6823,  Fax:  +1-313-577-8789.  E-mail:  kang@feedback.eng.wayne.edu 
’  Associate  professor  (Corresponding  Author),Tel:  +1-313-577-3888,  Fax:+1-313.577-8789.  E-mail:  tan@tan.eng.wayne.edu 

Submitted  to:  Sixth  International  Conference  on  Recent  Advances  in  Structural  Dynamics,  Institute  of  Sound  and  Vibration, 
Southampton,  England,  July,  1997 


1291 


A. 

NOMENCLATURE 

Area  of  shaft  cross  section  [m^] 

do 

Diameter  of  shaft  cross  section  [m] 

C 

Generalized  coordinate  of  an  incident  wave  [m] 

Cdt  {Ct) 

Translational  damping  coefficient  [N-sec/m] 

Cdr  (Cr) 

Rotational  damping  coefficient  [N-m-sec/rad] 

Co 

Bar  velocity  [m/sec] 

Cj 

Shear  velocity  [m/sec] 

D 

Generalized  coordinate  of  a  transmitted  wave  [m] 

E,G 

Young’s  and  shear  modulus  [NW],  respectively 

I 

Lateral  moment  of  inertia  of  shaft  [m'^] 

Jm  (Jm) 

mass  moment  of  inertia  of  a  rotor  mass  [kg-m"^] 

K 

Timoshenko  shear  coefficient 

Knikr) 

Rotational  spring  [N/rad] 

Kt  (kd 

Translational  spring  [N/m] 

t 

Length  of  shaft  [m] 

M{m) 

Mass  of  rotor  [kg] 

P 

Axial  force  [N] 

rij ,  ti] 

Reflection  and  transmission  coefficients,  respectively,  i  -  1  positive  traveling 

U{u) 

wave;  i  =  2  negative  traveling  wave;  jr  =  1  propagating  wave  for  Cases  11  and 
/V;  7  =  2  attenuating  wave  for  Cases  II  and  IV.  Both  7  =1 ,  2  for  propagating 
wave  for  Case  I 

Transverse  displacement  [m] 

X-Y-Zix-y-z) 

Reference  frame  coordinates  [m] 

a 

{K-G)IE 

Rotation  parameter,  see  Eqn.  (Id) 

£ 

Pl{E-A),  axial  strain 

e' 

Non-dimensional  axial  load  parameter,  see  Eqn.  (13b) 

T,  f  (r,  y) 

Wavenumber  [m‘‘] 

nAi) 

See  Eqns.  (20a,  b),  (22a,  b)  and  (24a,  b) 

p 

Mass  density  of  shaft  [kgW] 

G 

Diameter  ratio  between  two  shaft  elements 

57,  (ft)) 

System  natural  frequency  for  Timoshenko  model  [rad/sec] 

a 

System  natural  frequency  for  Euler-Bernoulli  model  [rad/sec] 

Q. 

Rotation  speed  of  shaft  [rad/sec] 

W  (¥) 

Bending  angle  of  the  shaft  cross-section  [rad] 

subscript  L,  r 

The  left  and  right  side  of  a  discontinuity,  respectively. 

superscript  -,+ 

Negative  and  positive  traveling  waves,  respectively,  when  used  in  C  and  D. 

Note:  Symbols  in 

Otherwise  denotes  quantities  on  the  left  and  right  side  of  a  discontinuity, 
respectively 

parenthesis  are  the  corresponding  non-dimensional  parameters. 

1292 

1.  INTRODUCTION 


The  vibrations  of  elastic  structures  such  as  strings,  beams,  and  plates  can  be  described  in 
terms  of  waves  propagating  and  attenuating  in  waveguides.  Although  the  subject  of  wave 
motions  has  been  considered  much  more  extensively  in  the  field  of  acoustics  in  fluids  and  solids 
than  mechanical  vibrations  of  elastic  structures,  wave  analysis  techniques  have  been  employed  to 
reveal  important,  physical  characteristics  associated  with  vibrations  of  structures.  One  advantage 
of  the  wave  technique  is  its  compact  and  systematic  approach  to  analyze  complex  structures  such 
as  trusses,  aircraft  panels  with  periodic  supports,  and  beams  on  multiple  supports  [1].  Previous 
works  based  on  wave  propagation  techniques  have  been  well  documented  in  several  books  [2-4], 
Recently,  Mead  [5]  applied  the  phase-closure  principle  to  determine  the  natural  frequencies  of 
Euler-Bemoulli  beam  models.  A  systematic  approach  including  both  the  propagating  and  near- 
field  waves  was  employed  to  study  the  free  vibrations  of  Euler-Bemoulli  beams  [6]. 

High  speed  rotating  shafts  are  commonly  employed  in  precision  manufacturing  and  power 
transmission.  Despite  the  usefulness  of  the  wave  propagation  method  in  structural  vibrations, 
applications  of  this  technique  to  study  the  dynamics  and  vibrations  of  a  flexible  shaft  rotating 
about  its  longitudinal  axis  have  seldom  been  considered.  The  purpose  of  this  paper  is  to  examine 
the  wave  reflection  and  transmission  [6]  in  an  axially  strained,  rotating  Timoshenko  shaft  under 
various  support  and  boundary  conditions.  The  effect  of  the  axial  load  is  included  by  considering 
the  axial  static  deformations  in  the  equations  of  motion.  This  paper  is  a  sequel  to  another  paper 
in  which  the  authors  discuss  the  basic  wave  motions  in  the  infinitely  long  shaft  model  [7]. 

Although  there  have  been  numerous  studies  on  the  dynamics  and  vibration  of  rotating  shafts, 
none  has  examined  the  effects  of  axial  strains  (which  cannot  be  neglected  in  many  applications) 
on  the  vibration  characteristics  of  a  Timoshenko  shaft  under  multiple  supports.  Modal  analysis 
technique  has  been  applied  to  study  the  vibration  of  a  rotating  Timoshenko  shaft  with  general 
boundary  conditions  [8,  9],  and  subject  to  a  moving  load  [10].  Recently,  the  distributed  transfer 
function  method  was  applied  to  a  rotating  shaft  system  with  multiple,  geometric  discontinuities 
[11],  The  wave  propagation  in  a  rotating  Timoshenko  shaft  was  considered  in  Ref.  [12].  Other 
major  works  on  the  dynamics  of  rotating  shafts  have  been  well  documented  in  Refs.  [13-15]. 

This  manuscript  is  organized  as  follows.  Governing  equations  of  motion  [16]  and  basic  wave 
solutions  for  the  Timoshenko  shaft  are  outlined  in  Section  2.  Each  wave  solution  consists  of  four 
wave  components:  positive  and  negative,  propagating  and  attenuating  waves.  In  Section  3,  the 
wave  reflection  and  transmission  matrices  are  derived  for  the  shaft  under  various  point  supports 
and  boundary  conditions.  The  supports  may  include  translational  and  rotational  springs  and 
dampers,  and  rotor  mass.  Results  are  presented  for  both  the  Timoshenko  and  the  simple  Euler- 
Bemoulli  models  to  assess  the  effects  of  axial  strain,  shaft  rotation,  shear  and  rotary  inertia.  The 
wave  propagation  across  a  shaft  with  geometric  discontinuities  such  as  a  change  in  the  cross- 
section  is  examined  in  Section  4,  and  the  wave  reflection  at  a  boundary  with  arbitrary  support 
conditions  is  considered  in  Section  5. 

With  the  wave  reflection  and  transmission  matrices  as  the  main  analytical  tools,  it  is  shown  in 
Section  6  how  to  apply  the  current  results  together  with  the  transfer  matrix  method  to  analyze  the 


1293 


free  vibration  of  a  rotating,  multi-span  Timoshenko  shaft  system  in  a  systematic  manner.  The 
proposed  approach  is  then  demonstrated  by  considering  the  free  vibration  of  a  two-span  beam 
with  an  intermediate  support. 


2.  FOKMULATION  AND  WAVE  SOLUTIONS 


Consider  a  rotating  shaft  subjected  to  axial  loads  and  with  multiple  intermediate  supports  and 
arbitrary  boundary  conditions,  as  shown  in  Fig.  1.  Including  the  effects  of  rotary  inertia,  shear 
deformations,  and  axial  deformations  due  to  the  axial  loads,  the  uncoupled  equations  of  motion 
governing  the  transverse  displacement  u  and  the  slope  \}/  due  to  bending  can  be  derived  in  the 
following  non-dimensional  form 


..  .  d'^u  „  d^u 

£ 

+16a(l  +  £)(l-i-e - )^r^  =  0  , 

a  dr 


U  lA  £  .  d~U 

■2ri3^  +  a— -16£{1  +  £--)^^ 


a  dz'^ 


-l-16a(l-l-£)(l  +  £-— =  0  , 
a  dr 


u  =  —  z  =  —  r  =  —  T  =  ^  ■ 
a.'  a.'  T.  ^  \  KG 


Figure  1.  A  rotating  Timoshenko  shaft  model  subject  to  axial  loads  and  with  general  boundary  conditions. 


1294 


(Id) 


a 


KG 
E  ’ 


E 


Note  that  u  and  y/are  the  measurements  in  the  complex  plane,  that  is  u=ux+iuy  and  \i/=\}fx+iYr 
E  denotes  the  Young’s  modulus,  p  the  mass  density,  As  the  area  of  the  cross  section,  ao  the 
diameter  of  shaft,  K  the  Timoshenko  shear  coefficient,  G  the  shear  modulus  and  Q  the  constant 
angular  velocity  of  the  shaft.  Details  of  deriving  these  equations  of  motion  are  found  in  Ref. 
[16]. 

Assuming  and  substituting  the  following  wave  solutions  into  Eqns.  (la)  and  (lb) 

=  (2a) 

=  (2b) 

and  defining  the  non-dimensionalized  wavenumber  7  and  system  natural  frequency  w  gives  the 
frequency  equation,  Eqn.  (3a);  see  Ref.  [7], 


y  =  r^o  . 


_  CO  a, 


KG  . 


(j)  = - a.  =  - is  known  as  the  shear  velocity) . 


where, 


7“*  -A7"  4-5  =  0  , 


A  =  (1  -i-  a)a)  ^  -  2j3  cu  “  16£  (1  +  e - ) , 


5  =  a  ot)^  -  2j3  u)  -  16a  (1 4- £)(1 4- e  -  •^) 


The  four  roots  of  Eqn.  (3a)  are 


7  =  ±-^[a  ±  -x/a^  -45  . 


(2c) 

(2d) 

(3a) 

(3b) 

(3c) 

(4) 


In  general,  7  is  complex.  Let  (o  be  real.  It  can  be  shown  that,  with  a  >  0  and  £  the  axial  strain 

of  the  elastic  solid,  the  discriminant  A^  -45  is  positive  semi-definite  for  most  engineering 
applications.  Hence,  it  is  possible  to  classify  the  wave  solutions  into  four  distinct  cases.  Note 
that  one  may  study  the  wave  propagation  by  considering  only  a  single  general  form  of  the  wave 
solution.  However,  the  classification  procedure  identifies  the  coupled  modes  of  vibration  of  the 


1295 


Timoshenko  shaft  model  and  provides  a  better  understanding  on  how  each  wave  solution  governs 
the  wave  motions  [7].  Based  on  the  algebraic  relationships  between  A  and  B,  the  four  valid  wave 


solutions  are  obtained  as  follows. 

Case  /  ( A  >  0  and  B  >  0 ); 

«(z,  t)  =  +  C- (5a) 

V(z,  t)  =  (CJ.e-'*’"  +  )e®  (5b) 

Case  !I{A>0  andB<0): 

«(z,  t)  =  (Ce-"'''  +  +  C:^e-^'-‘  +  (6a) 

yz(z,t)  =  (c;,^-'^"  -  c;,/"  +  +  0;^/=-  )£®  (6b) 

CaseIIIiA<0mdB>0): 

uiz,t)  =  (C>-'''  +  +  C:,«")«'“  (7a) 

Vr(z,0  =  (C;,a-"  +  C;,ef>'  +  +  C;je'=')«®  (7b) 

Coje/V(A<0  andB<0): 

k(zA)  =  (C,>-^'^  +C,>’‘'-  +  C>-'''=‘  +  C,V'’'*')«'®  (8a) 

V/(z,«)  =  (C;,e-f-  +  (8b) 

where, 

f,  =^(|A|  +  V-5’-4|B|f,  r.  (9^.  W 

r,  =  +  4|B|  +  |A|)^ ,  r,  +  4|B|  -  |A|j' ,  (9c,d) 


and  the  coefficients  C*  and  C"  denote  positive-  and  negative-travelling  waves  from  the  origin  of 
disturbance,  respectively.  Important  remarks  on  the  basic  wave  propagation  characteristics  are 
summarized  from  [7].  First,  the  wave  solution  of  Case  III  does  not  exist  in  the  real  frequency 
space  since  this  type  of  solution  represents  a  situation  in  which  none  of  the  wave  components  can 
propagate  along  the  waveguide.  Therefore  the  study  of  Case  III  is  excluded  in  the  present  paper. 
Second,  the  vibrating  motion  of  the  shaft  model  in  Case  I  is  predominately  pure  shear  [17]  which 


1296 


is  unique  for  the  Timoshenko  shaft  model,  while  in  Case  U  and  Case  IV  the  flexural  mode  and 
the  simple  shearing  mode,  which  are  corrected  by  including  the  rotary  inertia  and  shearing  effects 
in  the  formulation,  dominate.  Third,  when  the  shaft  rotates  at  a  very  high  speed  and/or  the  shaft 
is  axially  strained  by  tensile  loads,  the  wave  solution  of  Case  IV  governs  the  vibrating  motion  of 
the  shaft  model  in  the  low  frequency  range. 


For  comparison,  the  parameters  A  and  B  in  the  simple  Euler-Bernoulli  beam  model  are 


(10a) 

(10b) 

where,  the  non-dimensionalized  natural  frequency  6)  is  defined  as 


is  known  as  the  bar  velocity). 


(10c) 


Note  that,  because  B  is  negative,  wave  solutions  of  Case  I  and  Case  III  do  not  exist. 

In  general  the  displacement  and  the  rotation  of  an  infinitesimal  shaft  element  consist  of  four 
wave  components  as  shown  by  Eqns.  (5a-8b).  Once  the  displacement  and  the  bending  slope  are 
known,  the  moment  M  and  shear  force  V  at  a  cross  section  can  be  determined  from 


M  =  EI^, 
dz 

(ll) 

(du  ^ 

y  = J . 

(12) 

Moreover,  the  kinematic  relationship  between  the  transverse  displacement  and  the  slope  due  to 
bending  is 

S^u  3^u  .  ,3  r 

3r~  3z^  3z  ' 

(I3a) 

where  e'  denotes  the  effects  of  the  axial  force  and  is  defined  as 

e  =  l  +  e  — . 

a 

(I3b) 

1297 


3.  WAVE  REFLECTION  AND  TRANSMISSION  AT  SUPPORTS 


When  a  wave  is  incident  upon  a  discontinuity,  it  is  transmitted  and  reflected  at  different  rates 
depending  on  the  properties  of  the  discontinuity.  Consider  a  rotating  Timoshenko  shaft  model 
supported  at  ^  =  0 ;  see  Fig.  2.  The  support  simulates  a  bearing  modelled  by  linear,  translational 
and  rotational  springs,  dampers,  and  a  rotor  mass  which  typically  represents  a  gear  transmitting  a 
torque.  Based  on  Eqns.  (5a-8b),  group  the  four  wave  components  into  2x1  vectors  of  positive¬ 
travelling  waves  and  negative-travelling  waves  C" ,  i.  e.. 


Recall  that,  depending  on  the  system  parameters,  the  rotating  Timoshenko  shaft  model  has  four 
(practically  three)  different  wave  solutions  in  the  entire  frequency  region  as  described  in  Eqns. 
(5a-8b).  Thus  C^  and  C,  in  the  above  expression  do  not  always  correspond  to  propagating  and 

attenuating  wave  components,  respectively.  When  a  set  of  positive-travelling  waves  is 
incident  upon  the  support,  it  gives  rise  to  a  set  of  reflected  waves  C~  and  transmitted  waves  . 
These  waves  are  related  by 


C'=rC" 

(15) 

D"  =  tC", 

(16) 

where  r  and  t  are  the  2x2  reflection  and  transmission  matrices  respectively  and  are  expressed  as 


r  = 

'^u 

^12* 

(17) 

/2! 

''22. 

’^11 

^12 

t  = 

(18) 

/2I 

^22  _ 

From  Eqns.  (5a-8b),  suppressing  term  and  excluding  Case  III,  the  displacements  u~  and 
and  the  bending  slopes  y/~and  y/'^  at  the  left  and  right  of  z  =  0,  respectively,  can  be  expressed 
in  terms  of  the  wave  amplitudes  of  the  displacement.  For  convenience,  the  over-bar  (•)  on  the 
wavenumbers  is  dropped  hereafter. 


Case  1  {A  >0  andB>0); 

«-(z)  =  C, +  c;,£'^==,  (19a) 

W'U)  =  -n,C:,e-‘^“  -77, c;, +T7,C>-'>''- (19b) 


1298 


M.Jm 


Figure  2.  Wave  motion  at  a  general  support  (the  disk  may  be  considered  as  a  gear  transmitting  a  torque). 


where, 

_  72-®' 

T/i  —  /  »  ~  /  • 

r,£  72^ 

//  ( A  >  0  and  B  <  0 ): 

«-(z)  =  c;, +  C'e"'''  +  , 

i^-(z)  =  7,, C, >-'"■=  -77,C,-^r''''=  +)7,C*e-''“  -TJ,C,;;e"=S 


k*(z)  =  A>“’'"  +  AV’‘. 


where 


= 


r,£' 


>  Ba  “ 


CaselV{A<0a.ndB<0): 


(19c) 

(19d) 

(20a,  b) 

(21a) 

(21b) 

(21c) 

(21d) 

(22a,  b) 


1299 


the  following  set  of  matrix  equations  can  be  established  for  each  Case. 
Case  /  ( A  >  0  and  5  >  0 ): 


n  n 

r  1 

1  1 

.  ri  n 

c^  + 

rC^=: 

L^i  12] 

-^2j 

-im,  Ic+r  IrC" 

KYi-tIi)  i(r2“7?2)J  [-KYi-Vi)  -i(.Y2-Tl2)\ 


(28a) 


ri,(k,-J„,co^)  +  iTi,{c,0)~r,)  0)")  + 1772 72 )  ^28b) 

ik^~mO)-)  +  i{c,co  +  r^-r]^)  (fc, -mtt>")  +  z’(c,®  +  72 -772)  J 


1300 


Case  //  ( A  >  0  and  S  <  0 ): 


1 

.^1 


1 


1 


L^i 


1 

^2 


tc^ 


1 

1 

n 

-ir,T7, 

[i(r, -T7i)  r2-j772_ 

C  -r 

-^r, -77,) 

~  (r,  -  it], )_ 

(29a) 


r},(k^-J„^0)-)  +  irij  (c^co-r^) 
(k,  -mco^)  +  i(c,Q)  +  r,-T],) 


Tl2(k,-J„,co^-r2)  +  iV2CrO} 

{kj  -m(0^  +r2)+/(c,Ct}-  772) 


tc", 


(29b) 


Case  /y(A<0and5<0); 


'1  11 

■  1  1  ■ 

rC"  = 

'  1  r 

Jli  Hi. 

-^2  -^1. 

jii  ^1- 

■-1X2772  -£t,77,  ■ 

C'* 

-1X2772 

KT.-ni)  r, -i77,_ 

-r 

—  i(r2  —  772 ) 

-(r,  -  in,). 

(30a) 


T]2ik^-J„^(0'")  +  iT]^{c,C0-y2) 
{k,  -m(D‘)  +  i(c,aj  +  r2  -772) 


riiik,-J„co^-y,)  +  iT],c^co' 
{k,  - mco^  +  r, )  +  i{c,(0 -rij) 


tC", 


(30b) 


where  Eqns.  (15)  and  (16)  have  been  applied  in  all  Cases.  Note  that  in  Eqn.  (27a),  it  is  assumed 
that  the  rotational  spring  at  the  support  is  attached  to  the  cross  section  of  a  shaft  element  such 
that  the  rotational  spring  responds  only  to  the  slope  change  due  to  rotation  of  the  cross  section 
and  not  the  total  slope  change  of  the  neutral  axis  of  the  shaft  model.  This  assumption  allows  the 
shearing  motion  of  the  shaft  element  at  the  support.  Note  also  that  the  effect  of  axial  loads  on  the 
shear  force  at  the  support  is  neglected  since  the  contribution  of  axial  loads  to  the  shear  force  at 
the  support  or  boundary  is  small  compared  to  the  shear  force  due  to  the  flexural  motion  of  the 
shaft  element.  Exact  moment  and  force  balance  conditions  at  boundaries  for  a  rotating 
Timoshenko  shaft  element  subjected  to  axial  loads  can  be  found  in  Ref.  [16]. 

The  corresponding  matrix  equations  for  the  simple  Euler-Bernoulli  shaft  model  are  shown  in 
Appendix  I.  Solving  the  set  of  matrix  equations  simultaneously  for  r  and  t  gives  the  elements  of 
the  reflection  and  transmission  matrices  for  each  Case.  The  general  forms  of  solutions  to  these 
sets  of  equations  for  each  Case  is  not  presented  in  this  paper  due  to  space  limitation.  However 
one  can  obtain  the  solutions  in  either  closed-form  or  numerically.  Note  that  in  Case  II  and  Case 
IV,  the  first  columns  of  r  and  t  are  the  reflection  and  transmission  coefficients  due  to  incident 
propagating  wave  components,  and  the  second  columns  are  due  to  an  incident  attenuating  wave 
component  which  is  generally  termed  as  near-field  since  this  type  of  wave  decays  exponentially 
with  distance.  When  the  distance  between  the  origin  of  disturbance  and  the  discontinuity  is  very 


1301 


large,  these  attenuating  wave  components  can  be  neglected.  However,  as  mentioned  by  many 
authors,  for  example  Graff  [2],  attenuating  waves  play  an  important  role  in  wave  motions  by 
contributing  a  significant  amount  of  energy  to  the  propagating  wave  components  when  a  set  of 
propagating  and  attenuating  waves  are  incident  at  a  discontinuity  and,  in  particular,  when  the 
distances  between  the  discontinuities  are  relatively  small,  as  in  the  case  of  closely-spaced  multi¬ 
span  beams.  In  this  paper,  near-field  components  are  included.  In  what  follows,  the  effects  of 
the  point  supports  on  the  reflection  and  transmission  of  an  incident  wave  are  studied.  For 
comparison,  the  results  are  obtained  for  both  the  Timoshenko  and  the  simple  Euler-Bemoulli 
models,  which  hereafter,  for  brevity,  are  denoted  by  TM  and  EB,  respectively.  The  system 
parameters  used  in  the  numerical  results  are  taken  from  Ref.  [10];  ao  -  0.0955  m,  p  =  7700 
kgW,  K  =  0.9,  E  =  207x10^  Wnf,  G  =  77.7x10^  N/ml 


3.1.  Wave  reflection  and  transmission  at  rigid  supports 

Consider  two  cases:  the  simple  support  and  the  clamped  support.  The  r  and  t  are  solved  and 
shown  as  follows. 

•  Simple  support  (k,  =  oo,  =  m=  c,  =  -  7„,  =0) 

Case  7  ( A  >  0  and  B  >  0 ): 


r  = 


_ 1 _ 

(72 -7i)(r  1/2+6;') 


7i(6>"-72) 
72(7? -6;-) 


7i(6;’-72) 


(31a) 


t  = 


_ 1 _ 

(72-7i)(7i72+6)-) 


72(7?-®') 


Case  77  ( A  >  0  and  S  <  0): 

_ 1 _ 


_ 1 _ 

(iT,-r,xr,r,-!V) 


Case  7V(A<0andB<0): 


_ 1 _ 


7i(6)"“72) 
7i(72 -6)') 


r,(r,H®-)  1 


r,{r^  +  co^)' 


(31b) 


(32a) 


(32b) 


(33a) 


1302 


The  corresponding  reflection  and  transmission  matrices  for  the  EB  model  are  listed  in  App.  I 

Figures  3  and  4  plot  the  moduli  (magnitudes)  of  the  reflection  and  transmission  coefficients 
for  the  simple  and  clamped  supports.  The  finite  cutoff  frequencies,  above  which  all  waves 
propagate,  are  also  marked  in  the  figures.  Thus,  for  the  TM  model,  the  wave  motions  change 
from  Case  11  to  Case  1  when  (0>C0^  {0)^-4-  in  Fig.  3,  co,  =  4.24  in  Fig.  4;  O),  is  slightly  altered 
by  rotation  speed  and  axial  load).  The  results  show  that,  at  low  frequencies  (cD  <  0.1  =  3156 
rad/sec),  the  wave  reflection  and  transmission  coefficients  of  the  TM  model  agree  well  with  those 
of  the  EB  model  for  both  support  conditions.  However,  as  the  frequency  increases,  the  wave 
propagation  characteristics  of  the  TM  model  differ  significantly  from  those  of  the  EB  model. 
These  differences  can  be  explained  by  examining  the  different  modes  of  vibration.  When 
co>co^  (in  the  regime  of  Case  1),  the  vibrating  motion  of  the  TM  model  is  dominated  by  the  pure 
shearing  motion  [7,  17],  and  hence  the  EB  model,  which  neglects  the  rotary  inertia  and  pure 
shear  effects,  become  inaccurate  at  high  frequency.  As  discussed  in  Ref.  [7],  at  the  finite  cutoff 
frequency,  the  TM  shaft  experiences  no  transverse  displacement,  and  the  cross-section  of  the 
shaft  simply  rotates  back  and  forth  in  unison 

In  Figs.  3(d)-(f)  and  4(d)-(f),  for  =  0  and  £  =  0 ,  the  reflection  and  transmission  coefficients 
of  the  EB  model  are  independent  of  the  frequency.  This  is  because  from  Eqn.  (10a),  A  =  0,  and 
Eqns.  (9c, d)  lead  to  a  single  wavenumber  F,  =  Tj .  From  Appendix  I,  Eqns.  (32*-36*),  the  r  and 
t  are  thus  constant  matrices.  It  is  also  seen  that  the  wave  reflection  and  transmission  coefficients 


1303 


for  both  shaft  models  are  basically  independent  of  the  rotation  speed  over  the  entire  frequency 
range,  even  at  high  rotation  speed  =  0.05  =  44,600  rpm.  In  Ref.  [7],  it  is  also  found  that  has 
negligible  effects  on  the  system  frequency  spectrum,  phase  velocity  and  group  velocity.  On  the 
other  hand,  the  effects  of  the  axial  load  are  significant  for  both  propagating  and  attenuating 
waves  in  the  regime  of  Case  H,  see  Figs.  3(b)-(c)  and  4(b)-(c).  For  both  shaft  models  under 
simple  support  and  compressive  loads  (Figs.  3(b,  e)),  the  reflection  coefficient  ru  of  the  incident 
propagating  wave  is  reduced  significantly  in  the  regime  of  Case  II,  while  the  transmission 
coefficient  fj,  of  the  propagating  wave  component  increases  to  balance  the  energy  carried  in  the 
wave.  However,  the  attenuating  wave  component  which  does  not  carry  any  energy  loses  its 
transmissibility  in  the  same  amount  as  the  reflection  coefficient  r,2 .  Thus,  in  the  presence  of  a 
compressive  load,  most  of  the  transmitted  wave  energy  in  Case  II  comes  from  the  propagating 
component  of  the  incident  wave.  Note  that  axial  tensile  loads  have  the  reverse  effects  on  these 
wave  components.  In  the  clamped  support  case,  the  positive  propagating  wave  component  rn  is 
constant  over  the  regime  of  Case  II  under  any  loading  conditions  for  the  both  shaft  models,  as 
seen  in  Fig.  4. 

Since  there  is  no  damping  at  the  support,  the  incident  power  (Ilinc),  reflected  power  (Hrefi)  and 
transmitted  power  (Iltnin)  in  Cases  II  and  IV  are  related  by  flinc  =  rirefl+ritran  =  Hinc, 

or  Iriil^+Uiil^  =  1.  This  relationship  is  confirmed  by  the  plots  shown  in  Figs.  3  and  4,  where  for 
both  shaft  models,  Irni  and  knl  cannot  exceed  one.  However  in  the  regime  of  Case  I,  in  which 
all  wave  components  propagate,  the  energy  balance  is  Hinc  =  (l^ii+r2iP+lfii+r2!p)  Ilinc,  or  Hinc  = 
(Iri2+r22l^+l?i2+r22l^)  Hinc-  Together  with  the  plots  on  the  phase  of  these  coefficients  (not  shown 
to  minimize  the  size  of  this  manuscript),  the  above  relationships  can  also  be  verified  for  wave 
motion  of  Case  I. 


1304 


support  without  “resistance”.  The  impedance  mismatching  (rn  =  1,  ?,,  =  0)  frequency  at  which 
the  propagating  wave  component  is  completely  reflected  without  being  transmitted  can  also  be 
determined  from  Figs.  4(b,  d)  for  the  two  shaft  models.  This  impedance  mismatching  frequency 
is  located  in  the  regime  of  Case  II  for  the  TM  model  where  the  transverse  mode  dominates  the 
vibrating  motion  of  the  shaft.  Numerical  results  show  that,  as  the  spring  constant  increases,  this 
impedance  mismatching  frequency  increases,  but  is  limited  to  within  the  regime  of  Case  II  and 
can  never  be  found  in  the  regime  of  Case  I  where  the  pure  shearing  mode  dominates  the  vibrating 
motion  of  the  shaft  (refer  to  Fig.  3  for  the  transition  of  types  of  wave  motion). 

Figure  6  shows  the  reflection  and  transmission  coefficients  for  waves  incident  upon  a  support 
having  both  translational  and  rotational  constraints.  Since  both  flexural  and  shearing  modes  of 
vibration  are  constrained  at  this  support,  the  maximum  of  the  reflection  coefficient  is  expected  to 
be  higher  than  the  previous  case.  Figures  6(a-b)  and  (c-d)  are  the  results  for  the  TM  and  EB 
models,  respectively.  The  translational  and  rotational  spring  constants  used  in  the  simulations 
are  k,Q-  10^  N/m  and  ^^5=  10^  Nm/rad,  respectively.  It  is  noted  that  in  the  regime  of  Case  II,  i.e.. 


Figure  6.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  translational  and  rotational  springs 
{k,-k,Q,  k^^O,  c,  =  c,  =  m=  y,„  =  0)  as  a  function  of  frequency,  J3  =  0.05  and  e  =  0.  (a-b)  and  (c-d)  are  results 
for  the  Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


modulus  modulus 


2.0 


r,2  (upper  line) 

r,2=:r2,  (middle  dashed  line) 

Tj,  (lower  line) 


Figure  4.  Wave  reflection  coefficients  at  a  clamped  support  (/c,  =  fc,  =  ■»  and  =  c,  =  m  =  =  0)  as  a  function  of 

frequency,  (a)-(c)  and  (d)-(f)  are  the  results  for  the  Timoshenko  and  Euler-BernouUi  shaft  models,  respectively.  The 
transition  from  one  type  of  ivave  motion  to  another  is  marked  for  the  case  /J  =  0.05 ,  £  =  -0.05 . 


3.2,  Wave  reflection  and  transmission  at  elastic  supports 

Figure  5  shows  the  reflection  and  transmission  coefficients  for  waves  incident  upon  a  support 
with  a  finite  translational  spring  for  three  different  spring  constants.  Figures  5(a)-(b)  and  (c)-(d) 
are  results  for  the  TM  and  EB  models,  respectively.  The  spring  constant  used,  k^Q  =  10^  N/m,  is 
a  typical  bearing  spring  constant  value  for  turbine  generators.  The  plots  show  that  there  is  no 
significant  difference  in  the  moduli  between  the  two  shaft  models.  This  is  because  the  incident 
wave  does  not  experience  any  rotational  constraint  at  the  support,  and  hence  the  additional  rotary 
inertia  factor  in  the  TM  model  has  only  a  small  contribution  to  the  wave  motions.  As  the  support 
spring  constant  increases,  the  curves  for  both  the  reflection  and  transmission  coefficients  are 
shifted  to  the  right  and,  as  the  spring  constant  approaches  infinity,  these  curves  eventually 
become  asymptotic  to  those  shown  in  Fig.  3.  Note  that  an  impedance  matching  (r  =  0,  t  =  I), 
where  all  wave  components  are  transmitted  without  being  reflected,  is  found  in  the  high 
frequency  region  for  both  shaft  models.  Thus,  as  the  frequency  increases,  the  characteristics  of 
waves  travelling  along  the  shaft  remain  unchanged  such  that  waves  propagate  through  this  elastic 


Figure  5.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  a  translational  spring 

=  c,  =  c,  =m  =  =0)  as  a  function  of  frequency,  /3  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 

Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


1308 


in  the  low  frequency  range,  both  shaft  models  have  similar  reflection  characteristics,  and  both  the 
reflection  and  transmission  coefficients  are  not  significantly  affected  by  the  rotational  spring. 
However,  as  the  frequency  increases,  the  effect  of  the  rotational  constraint  on  the  wave  motion 
becomes  eminent,  particularly  for  the  TM  model.  As  seen  in  Figs.  6(a)  and  6(c),  the  reflection  of 
the  attenuating  wave  components  are  significantly  higher  than  those  of  the  propagating  wave 
components.  Hence,  when  a  rotating  shaft  has  a  clamped  support(s)  such  as  a  journal  bearing, 
contributions  from  the  attenuating  wave  components  should  be  included  in  the  formulation  since 
a  significant  amount  of  energy  in  the  propagating  component  arises  from  the  incident  attenuating 
wave  component.  It  is  noted  that  the  impedance  matching  regions  seen  in  Figs  5(a,  c)  disappear 
when  the  rotational  constraint  is  added.  Moreover,  the  impedance  mismatching  frequency  shown 
in  Figs.  5(b,  d),  which  is  found  in  the  regime  of  Case  II,  also  does  not  occur.  At  low  frequency  in 
Figs.  6  (b,  d),  there  appears  to  be  a  mismatching  region,  but  rj,  is  not  exactly  equal  to  zero. 
From  Figs.  6(a,  c),  it  is  seen  that  there  is  a  frequency  at  which  the  positive  propagating  wave 
component  r,,  is  zero  (this  frequency  is  slightly  different  for  the  two  models).  This  frequency 
does  not  correspond  to  an  impedance  matching,  though  the  propagating  wave  is  not  reflected  at 
all  but  is  only  transmitted  (r,i  =  1).  Based  on  other  research  results  [18],  this  phenomenon  likely 
indicates  a  structural  mode  delocalization  in  bi-coupled  systems,  in  which  vibrations  on  both 
sides  of  the  support  become  strongly  coupled.  Further  research  on  the  vibrations  of  rotating 
shafts  with  intermediate  supports  is  being  pursued  to  confirm  the  mode  delocalization. 

Figure  7  plots  the  effects  of  axial  compressive  loads  on  the  wave  reflection  and  transmission 
upon  a  support  with  finite  spring  constant  for  the  Timoshenko  shaft  model.  As  seen  in  Fig.  7(a), 
the  reflection  coefficient  for  the  incident  propagating  wave  component  ru  is  substantially 
reduced  in  the  low  frequency  range  while  the  reflection  coefficient  for  the  incident  attenuating 
wave  component  increases  significantly.  However,  Fig.  7(b)  shows  the  reversed  effects  on  the 
transmission  coefficient.  It  can  therefore  be  concluded  that,  when  the  shaft  is  axially  strained  by 


Figure  7.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  (A:^  =  k,Q  and  kr  =  c,  =  Cr  =  m  =  J„,  =  0) 
for  the  Timoshenko  shaft  model  with  and  without  the  compressive  load,  (a)  reflection  coefficients,  (b)  transmission 
coefficients. 


1309 


Figure  8.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  k^  =  and  c,=  Cr  =  m  = 

J„,  =  0)  for  the  Timoshenko  shaft  model  with  and  without  the  compressive  load,  (a)  reflection  coefficients,  (b) 
transmission  coefficients. 


compressive  loads,  the  energy  contribution  from  the  incident  attenuating  wave  component  to  the 
energy  in  the  reflected  propagating  wave  is  more  significant  than  the  strain-free  situation  in  the 
low  frequency  range,  while  most  of  the  energy  in  the  transmitted  wave  derives  from  the  incident 
propagating  wave  component. 

Figure  8  plots  the  wave  reflection  and  transmission  coefficients  along  an  axially  compressed 
Timoshenko  shaft  model  at  a  support  with  finite  translational  and  rotational  spring  constants. 
Similar  results  to  the  previous  example  can  be  observed  in  terms  of  energy  contribution  from  the 
incident  attenuating  wave  component  in  the  low  frequency  range.  However,  the  effects  of  the 
axial  compressive  load  on  both  the  reflection  and  transmission  coefficients  for  the  propagating 
wave  component  ( r^^  and  r,, )  are  significantly  reduced  when  compared  to  Fig.  7. 


3.3.  Wave  reflection  and  transmission  at  damped  supports 

Figure  9  shows  the  effects  of  both  translational  and  rotational  dampers  at  a  support  with  finite 
translational  and  rotational  spring  constants.  Figures  9(a,  b)  and  (c,  d)  are  results  for  the  TM  and 
EB  models,  respectively.  The  translational  and  rotational  damping  constants  used  in  this  study 
are  c^to  =  2x10^  Ns/m  and  Cdro  =  64x10^  N-m-s/rad,  typical  values  for  bearings  in  turbine 
generators.  The  curves  with  symbols  (•  and  ♦)  are  the  results  when  the  rotational  damping  factor 
is  also  included  in  the  formulation.  It  can  be  seen  that  I  and  Ir^l  for  both  shaft  models  are 
significantly  lowered  due  to  the  presence  of  damping.  Note  that,  because  of  the  damping,  the 
frequency  at  which  1  rj,  I  =  0  (compare  with  Figs.  6(a,  c))  no  longer  exists  for  both  shaft  models.  It 
can  also  be  seen  that  the  effect  of  the  rotational  damping  factor  on  the  wave  reflection  and 
transmission  is  not  significant  over  the  entire  frequency  range  for  both  shaft  models.  For  TM 
model,  the  contribution  of  the  rotational  damping  to  both  Ir;,  1  and  1?,,  I  is  almost  negligible.  The 


1310 


support  condition  considered  in  this  particular  example  is  simulated  as  an  actual  bearing  support 
adopted  in  turbine  generators.  Hence  for  this  particular  type  of  bearing  support,  the  effect  of  the 
rotational  damping  on  wave  reflection  and  transmission  is  not  considerable.  Other  numerical 
results  (not  shown  in  this  paper)  show  that  the  wave  propagation  at  the  damped  support  is 
characterized  by  translational  damping  rather  than  rotational  damping.  Note  that  similar  results 
have  been  presented  for  the  support  without  damping  (see  Fig.  6). 


Figure  9.  Wave  reflection  and  transmission  coefficients  at  an  elastic  support  with  damping  (k,  =  k,o,  K  =  Ko,  c,  =  c,;,o, 
c,  =  cjro  and  m  =  J,„  =  0)  as  a  function  of  frequency  for  /3  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 
Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


3.4.  Wave  reflection  and  transmission  at  a  rotor  mass 

Consider  a  gear  rigidly  assembled  to  a  rotating  shaft.  The  gear  is  assumed  to  be  perfectly 
balanced  and  its  thickness  is  sufficiently  small  such  that  wave  reflection  and  transmission  due  to 
the  geometric  discontinuity  between  the  shaft  and  the  gear  can  be  neglected.  However  the  gear 
does  resist  the  translational  and  rotational  motions  of  the  cross-sectional  element  of  the  shaft. 


1311 


Figure  10  shows  the  reflection  and  transmission  upon  the  gear  when  the  mass  mo  and  mass 
moment  of  inertia  J^o  of  the  gear  are  4  and  16  times  of  the  shaft,  respectively.  Not  shown  in 
Figs.  10  (b,  d)  is  that  r,2  =  0  when  w=  /„  =  0.  Like  some  previous  support  conditions 

discussed,  the  effects  of  the  rotor  mass  are  much  more  significant  in  the  high  frequency  region 
for  both  models  (particularly  around  and  beyond  the  cutoff  frequency  for  the  TM  model).  In 
general,  the  rotor  mass  decreases  the  transmission  and  increases  the  reflection  of  the  wave.  At 
very  high  frequency,  there  is  basically  no  wave  transmission.  Note  that,  since  the  geometric 
discontinuity  between  the  shaft  and  the  gear  is  neglected  in  this  model,  one  may  expect  that  the 
actual  reflection  for  both  the  propagating  and  attenuating  wave  components  would  be  higher. 


(a) 


(m=  =  V 


(b) 


- (m  =  0, 

- (  m  =  mg .  ) 


/ 


Figure  10.  Wave  reflection  and  transmission  at  a  rotor  mass  assembled  to  a  rotating  shaft  (*,  =  0  =  L  =  Cf  =  Cr  =  0, 
and  m  =  rriQ,  and  J,„  =  J„^)  as  a  function  of  frequency  when  P  =  0.05  and  £  =  0.  (a-b)  and  (c-d)  are  results  for  the 
Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively. 


4.  WAVE  REFLECTION  AND  TRANSMISSION  AT  A  GEOMETRIC  DISCONTINUITY 
It  is  common  for  a  rotating  shaft  element  to  have  changes  in  cross-section,  or  to  be  joined  to 


1312 


Case  I,  II,  or  IV 

A,  =  (1  +  a)co-  -  2j3co  -  16£  (1  +  £  -  ~) 

a 


Z=0 


Case  I,  II,  or  IV 


B,  =  co' 


ccco-  -  2p(o  -  I6a  (1  +  £)(1  +  £ - ) 

a 


A={\  +  a)(o^-2^co-^^^  (l  +  £,  -— ) 
o"  a 

16a 


=  co^\ a  (O'  -215(0-^^  (l+£j(l  +  £  -— ) 

L  O’  a 


Figure  11.  Wave  reflection  and  transmission  at  a  geometric  discontinuity. 


another  shaft  element  by  a  coupling.  Figure  1 1  shows  a  typical  example  of  a  discontinuous  shaft 
model  in  which  two  shafts  of  differing  wavenumber  and  diameter  are  joined  at  z  =  0 .  The 
subscripts  I  and  r  denote  z  =  O'  and  z  =  0^  regions,  respectively.  It  is  known  that  when  a  wave 
encounters  a  junction  or  a  discontinuity,  its  wavenumber  is  changed.  It  is  therefore  possible  that 
a  wave  on  the  left  side  of  the  junction  can  be  propagating,  while  after  crossing  the  junction  to  the 
right  side,  the  wave  becomes  attenuating.  Therefore,  for  a  Timoshenko  shaft,  when  a  wave 
propagates  through  the  junction,  there  are  mathematically  nine  possible  different  combinations  of 
wave  motions  to  be  considered  depending  on  the  values  of  the  functions  A  and  B  on  each  side  of 
the  junction,  as  depicted  in  Fig.  12. 


Figure  12.  Nine  possible  combinations  of  wave  motions  at  a  geometric  discontinuity  of  the  cross  section  for  the 
Timoshenko  shaft  model.  Subscripts  /  and  r  denote  the  left  and  the  right  side  of  the  discontinuity,  respectively. 


1313 


For  simplicity,  assume  that  material  properties  such  as  p,  E,  and  G  are  the  same  for  both  sides 
of  shaft  element.  The  displacement  continuity,  moment  and  force  equilibrium  conditions  are 
applied  at  the  junction  to  determine  the  wave  refection  and  transmission  matrices.  Results  for  the 
three  most  commonly  encountered  possibilities  in  the  low  frequency  regime  are  listed  as  follows. 

Case  II  {A,  >0,  Bi  <0)  -  Case  I  (A^  >  0 ,  R,  >0): 


■  1  r 

1 

1  ' 

r  1  M 

rC"  = 

tCY 

Jlu  ^21 . 

nu  ■ 

~n2i. 

Jl\r  ^2r_ 

r  -i^Bu 

^21^2! 

C"  + 

r  Xtnu 

”Y,r72,  1 

L^xr., 

-riu)  r 

21 -in  21^ 

1 

Y 

rr 

1 

T 

^772,)J 

rC-^ 


Yuriu  72rn2r 

L-io-"(r„-77„)  -i<y^(r2r-V2r)J 


tc", 


Case  II  (A,  >0,  B,  <0)  -  Case  II  (A^  >  0 ,  B^<0): 


■  1  1  ■ 

■  1  1  ■ 

■  1  1  * 

C"  + 

rC^  = 

Jlv  n2l. 

—  nu  “^2/. 

Y.r  n2r. 

(37a) 


(37b) 


(38a) 


-"^21^21 

^2,  -in 21 


C"  + 


i^unu  ^21^21 

~i(^H  —  nu  )  ~(^2/  ~  ^^21  ) 


rC" 


-i<yXrnu  -^Xr'n2r 
-zcr"(r,,-77„)  -C7“(r2,-7]2,)J 


CaseII{A^>0,  B,  <0)  -  Case  IV  {A,  <0 ,  S,  <0): 


r  i  1  1 

r  1  1 1 

■  1  r 

C"  + 

rC^  = 

nu  n2i 

l-nu  -n2i^ 

n2r  n^r. 

Y/^2/ 

^2/^2/ 

JXi—nu)  Y/~^*^2;. 

“(Yf  -in2i)_ 

-i^%rn2r 


-<yXrn:r 


tc\ 


(38b) 


(39a) 


(39b) 


where  cr  is  the  diameter  ratio  between  the  shaft  elements,  defined  as 


1314 


(40) 


CJ  =  — . 


Note  that  77/5  in  Eqns.  (37a-39b)  are  given  by  Eqns.  (20a,  b),  (22a,  b),  and  (24a,  b)  according  to 
the  type  of  wave  motion,  and  77/  s  on  the  right  side  of  the  geometric  discontinuity  are  modified 
as  follows. 


where 


:  -  77  = - —  for  Case  I, 

r„e;  yzX 

(41a,  b) 

-co^  Tl+co- 

,  Ti,  -  — - 7-  for  Case  II, 

r„e;  iT^rS; 

(42a,  b) 

,7]2  for  Case  IV, 

ir,x  r,,e; 

(43a,  b) 

£'  =  1  +  8,.  — and  £,  =  —  . 

(44) 

a  O' 


Moreover,  the  wavenumbers,  A  and  B  of  the  shaft  element  on  the  right  side  of  the  junction  are 
modified  as  follows. 

y „  =  ^[a,  +  ^Aj-4B^f .  r,,  =  -  Va?-4B,)^  ,  (45a.  b) 

+  =  .  (45c, d) 

where, 

A.  =  (l  +  a)a.=  -2pa)-^a  +  £,-|), 


B.  =co‘ 


am--2Pa-^  (l  +  £,)(l  +  £,-^) 

C  OL 


(46b) 


Corresponding  results  for  the  simple  Euler-Bemoulli  shaft  model  are  listed  in  Appendix  II. 

Figures  13  to  16  show  some  representative  examples  of  wave  reflection  and  transmission 
upon  the  geometric  discontinuity.  In  Figs.  13  and  14,  the  thick  and  thin  curves  represent  results 


1315 


for  the  TM  and  EB  models,  respectively.  The  second  graph  in  each  figure  shows  the  changes  of 
Ai,  Bi,  Ar  and  Br,  and  how  wave  solutions  on  both  sides  of  the  discontinuity  change  as  the 
frequency  increases  for  the  TM  model.  In  general,  the  wave  reflection  and  transmission  for  the 
EB  model  are  frequency  independent  except  when  the  shaft  is  axially  strained,  while  the  wave 
propagation  characteristics  for  the  TM  model  are  strongly  dependent  on  the  frequency. 

Comparing  Figs.  13  and  14,  it  is  noted  that,  for  both  shaft  models,  the  average  reflection  and 
transmission  rates  for  cr  =  0.8  are  higher  than  those  for  o'  =  1.2 ,  especially  for  the  attenuating 
wave  components.  These  results  imply  that  incident  attenuating  waves  contribute  more  energy  to 
propagating  waves  at  the  discontinuity  when  the  waves  travel  from  a  smaller  to  a  larger  cross- 
section.  In  particular,  it  is  noted  that  the  transmissibiiity  of  the  attenuating  wave  tn  has  a  strong 
dependency  on  the  direction  of  propagation.  Note  also  that  the  differences  between  the  two  shaft 
models  are  more  pronounced  when  cr  =  0.8 .  It  is  clearly  seen  from  the  figures  that  when  Bj  and 
B^  change  from  negative  to  positive,  both  reflection  and  transmission  coefficients  experience  a 
sharp  jump  or  drop  at  the  finite  cutoff  frequencies,  due  to  changes  in  the  types  of  wave  motion. 
In  the  frequency  region  (S,  >  0  and  B^  <0)  located  between  the  two  cutoff  frequencies  in  Fig. 


2.0  h 


0.5 


Thick  curves  :  Timoshenko  shaft  model 
Thin  curves  :  Euier-Bernoulli  shaft  model 


- Ui 


h 

i 

il _ 


- 

7/  X. - V' 


.'7 


\ 


5 

(O 


Figure  13.  Reflection  and  transmission  of  waves  incident  upon  a  change  in  the  cross-section,  a=  0.8,  P  =  0.05,  and 
£  =  0.  Thick  and  thin  curves  are  results  for  the  Timoshenko  and  Euier-Bernoulli  shaft  models,  respectively.  Note 
that  the  ordinates  in  the  lower  graphs  keep  increasing  with  frequency  (abscissa). 


1316 


13,  the  wave  motion  on  the  left  side  of  the  junction  is  governed  by  the  wave  solution  of  Case  I 
since  all  wave  components  are  propagating  at  a  frequency  larger  than  the  cutoff  frequency,  while 
the  wave  motion  on  the  right  side  of  the  junction  is  governed  by  the  wave  solution  of  Case  II. 
Thus,  for  O’  =  0.8 ,  some  of  the  propagating  wave  components  on  the  left  side  of  the  shaft  element 
cannot  propagate  as  they  pass  the  discontinuity,  and  become  attenuating.  A  similar,  but  converse 
conclusion  can  be  drawn  for  the  frequency  region  (5^  >  0,  5,  <  0)  when  cr  =  1.2 ,  as  shown  in 
Fig.  14.  The  results  of  Figs.  13  and  14  show  that,  for  different  system  parameters  cr,  jS,  and  e  and 
at  any  given  frequency,  the  types  of  wave  motion  on  each  side  of  the  discontinuity  can  be 
different,  as  depicted  in  Fig.  12. 

From  Eqns.  (41a-43b),  it  is  seen  that  when  the  Timoshenko  shaft  is  axially  strained  and  (O  is 
not  sufficiently  large,  the  wavenumber  (hence  wave  propagation  characteristics)  depends  strongly 
on  the  cross-section  ratio  <7.  Figure  15  shows  the  effects  of  the  axial  load  on  the  wave  reflection 
and  transmission,  which  are  mostly  limited  to  the  relatively  low  frequency  region.  In  Figs.  15(a- 
b),  when  the  shaft  is  axially  compressed  (£  =  -0.05),  the  reflection  and  transmission  due  to  the 
incident  attenuating  wave  component  decrease  for  both  o  <  1  (plot  (a))  and  <7  >  1  (plot  (b)). 
However,  the  transmission  due  to  an  incident  propagating  wave  decreases  significantly  for 


—  —  f,  2 


2.0  h 


Figure  14.  Reflection  and  transmission  of  waves  incident  upon  a  change  in  the  cross-section,  cr=  1.2,  /3  =  0.05,  and 
£  =  0.  Thick  and  thin  curves  are  results  for  the  Timoshenko  and  Euler-Bernoulli  shaft  models,  respectively.  Note 
that  the  ordinates  in  the  lower  graphs  keep  increasing  with  frequency  (abscissa). 


1317 


Figure  15.  Reflection  and  transmission  of  waves  upon  a  change  in  the  cross-section  when  =  0.05  for  the 
Timoshenko  shaft  model,  (a)  <7=0.8  and  £  =  -0.05.  (b)  (T=  1.2  and  e= -0.05.  (c)  cr=  0.8  and  £=  0.05.  (d)  or  =  1.2 
and  e=  0.05.  Thin  and  thick  curves  show  the  results  when  the  shaft  is  strain-free  (£=  0)  and  strained,  respectively. 


c  -  0.8  and  increases  for  <7  =  1.2  at  low  frequency. 

Effects  of  the  axial  load  on  the  wave  reflection  and  transmission  are  more  significant  when 
the  shaft  is  compressed  (Figs.  15(a-b))  than  when  it  is  under  tension  (Figs.  15(c-d)).  This  is 
because  the  wavenumbers  of  both  the  propagating  and  attenuating  wave  components  are  only 
slightly  changed.  It  is  also  noted  that,  in  the  low  frequency  range,  the  wave  solution  of  Case  IV 
governs  the  wave  motions  on  both  sides  of  the  discontinuity,  and  the  wave  components  which 
have  large  wavenumber  (Fi)  attenuate,  while  wave  components  with  small  wavenumber  (r2) 
propagate  along  the  waveguide  as  long  as  A  remains  negative. 


1318 


5.  WAVE  REFLECTION  AT  BOUNDARIES 


When  a  wave  is  incident  upon  a  boundary,  it  is  only  reflected  because  no  waveguide  exists 
beyond  the  boundary.  Consider  an  arbitrary  boundary  condition  with  translational  and  rotational 
spring  constraints,  dampers,  and  a  rotor  mass,  as  shown  in  Fig.  16.  The  reflection  matrix  at  the 
boundary  is  derived  for  each  Case.  Applying  the  same  non-dimensional  parameters  employed  in 
Section  3,  and  by  imposing  the  force  and  moment  balances  at  the  boundary,  which  can  be 
deduced  by  eliminating  and  inEqns.  (27a,  b), 

M~  =  k^y/  +  c^yr  +  J„y/ ,  (47  a) 

-V  =  k,u  +  c,u  +  mu ,  (47b) 

the  reflection  matrix  for  each  Case  is  determined. 

CaseI{A>0,  B>0): 

J  r72(ir2-2:,J  T’r-T7,(jri  +  ^.)  + 

+  Kr2-Bi)  +  ^s\  Kr2-T72)-^.J’ 


Case  7/  ( A  >  0 ,  5  <  0): 

7],(ir,-2:j  T'r-77,(ir,+Ej  -7]2(r2  +  zj1 

'''[i(r,-r7,)+z,  (r,-ii7,)+i.J  [i(r,-j?,)-z, 


M,Jm 


Z=0 


Figure  16.  Wave  reflection  upon  a  general  boundary. 


1319 


CaseIV{A<0,  5<0): 


772  (^r2  s„,) 

iCFj  -  7)2)  + 


(r.-mj+s. 


772  (zTj  +  Sp,) 

/(r2  ~  ^2)  ■“ 


-^iCr.+zj 

(r,-j77,)-E, 


(50) 


where  77's  in  above  equations  have  been  defined  in  Eqns.  (20a,  b),  (22a,  b)  and  (24a,  b),  and 

Z„,  =  +  ic^co  -  J,„co~ ,  and  =  ^,  +  ic,Q)  -  mco^  ,  (51a,  b) 

The  corresponding  results  for  the  simple  Euler-Bernoulli  shaft  model  are  listed  in  the  Appendix 
in.  By  specifying  the  parameters  in  the  reflection  matrix  r,  results  for  three  typical  boundary 
conditions  (simple  support,  clamped  support,  and  free  end)  can  be  obtained. 

•  Simple  support  {k,  =^,  =m  =  c,  =  =  7,„  =  0) 


r 


-1  0 
0  -1 


for  Case  I,  II,  and  IV, 


(52) 


•  Clamped  support  (k^  =  k,.  =  m  =  c,  =  =  J„,  =  0) 

1 

Tli-ri2 


(53b) 

(53c) 

•  Free  end  {k,  =  k^  =  m  =  c,  =  =  /„,  =  0 ,  and  £  =  0) 


1 


rii+ri2 


2772 


77,-  772  L-277i  -(r?, +772)J 

1 


-irii+Tli)  "2772 

2t},  (771+772) 


for  Case  II, 


for  Case  IV, 


■77, +7?2  2772 

-277,  -(771+772), 


for  Case  I, 


(53a) 


r 


J_ 


77,7?2(ri+72)”7ir2(77, +772) 
-277,7i(77, -7i) 


277272(772-72) 

-77,772(71 +72)  +  7i72(77i +772). 


(54a) 


where.  A,  =  r],n,  -  7, )  +  7,72 (77,  -  ^2 )  for  Case  I, 


+72)  +  7i72(77i  +772)  2i7}^y^iin^ -y^) 

^  A;,  [  2/77,7,(771 -7,)  77,772(/7, +72)-7i72(7?, +772) 


(54b) 


where,  =77,772 (77,  -72) -7,72(7?,  - 772)  for  Case//, 


1320 


Figure  17.  An  example  of  a  rotating  shaft  with  multiple  supports  and  discontinuities. 


I  \TiiV2(ri+ir2)-rj2(Tii+r]2)  277,7,(77, +ir,)  '  ^ 

r  = -  (54c) 

A/v  L  -2i7?2r,(77j-7j  -77,772(7,  +J72)+rir2(^i  +^2). 

where,  A;^  =77,772(71  -172)  “7172 (Hi -772)  for  Case  IV. 

6.  APPLICATIONS 

The  reflection  and  transmission  matrices  for  waves  incident  upon  a  general  point  support  or  a 
change  in  cross-section  can  be  combined  with  the  transfer  matrix  method  to  analyze  the  free 
vibration  of  a  rotating  Timoshenko  shaft  with  multiple  supports  and  discontinuities,  and  general 
boundary  conditions.  The  basic  idea  of  this  technique  has  been  shown  in  Ref.  [6].  However,  due 
to  the  complex  wave  motions  in  the  Timoshenko  shaft  model,  such  as  the  frequency  dependency 
of  the  wave  reflection  and  transmission  at  a  cross-section  change,  it  is  important  to  apply  the 
proper  reflection  and  transmission  matrices  consistent  with  the  values  of  A  and  B  on  both  sides  of 
the  discontinuity,  particularly  when  numerical  calculations  are  performed.  Consider  for  example 
the  free  vibration  problem  of  the  rotating  Timoshenko  shaft  model  shown  in  Fig.  17.  Denoting  R 
as  a  reflection  matrix  which  relates  the  amplitudes  of  negative  and  positive  travelling  waves  at  a 
discontinuity,  and  defining  T/  as  the  field  transfer  matrix  which  relates  the  wave  amplitudes  by 

C^iZo  +  z)  =  TC"(Zo) ,  C-(zo  +  z)  =  T-'C-(zo) ,  (55) 

the  following  relations  can  be  found. 


1321 


W-=R5W^ 

(R5=>-s). 

(56a) 

i  =  2,3,4  (station  number), 

/i  =  left  (/)  or  right  (r) 

(56b) 

w>T,w-,, 

(56c) 

<  =r,w-, 

(56d) 

<=T.w;, 

(56e) 

where  in  Eqn.  (56b), 

R,,=TiR„,,T,.  R,,=r,+t,(R-’-r,-)-'t,  (56f) 


Solving  the  above  matrix  equations  gives 

(r,T,R„T,-I)w>0,  (57) 

where  each  element  of  the  matrix  is  a  function  of  two  different  wavenumbers  and  the  frequency 
CO.  For  non-trivial  solutions,  the  natural  frequencies  are  obtained  from  the  characteristic  equation 

Det[(r,T,R„T,-I)]  =  0.  (58) 

The  proposed  method  is  applied  to  an  example  of  a  two-span  rotating  shaft,  simply  supported 
at  the  ends  and  with  an  intermediate  support  consisting  of  translational  and  rotational  springs,  as 
shown  in  Fig.  18.  Numerical  computations  were  performed  by  a  PC-based  Mathematical.  The 
values  of  the  spring  constants  Kj  and  Kr  are  those  introduced  in  Section  3.2,  with  ^  =  1  m  and 
the  rotation  parameter  p  =  0.05 . 


1322 


Figure  19  shows  the  first  eight  natural  frequencies  of  the  vibrating  shaft  for  both  the  TM  and 
EB  models  for  a  classical  simple  intermediate  fixed  support  K^=0)  placed  at 

various  locations.  The  results  confirm  the  well-known  fact  that  the  Timoshenko  model  leads  to 
smaller  eigenvalues.  Figure  20  shows  the  first  eight  natural  frequencies  of  the  Timoshenko  shaft 
for  an  elastic  intermediate  support  with  three  different  translational  spring  constants.  It  can  be 
seen  that  the  effect  of  the  translational  spring  diminishes  for  higher  modes.  The  proposed  wave 
analysis  technique  can  also  be  applied  effectively  to  the  study  of  structural  mode  localizations  in 
mistuned,  rotating  systems.  Dynamics  of  such  systems  will  be  addressed  in  another  paper. 


Figure  19.  Natural  frequencies  of  a  two-span,  rotating 
shaft  as  a  function  of  the  support  location;  intermediate 
support  is  fixed. 


Figure  20.  Natural  frequencies  of  a  two-span  rotating 
Timoshenko  shaft  as  a  function  of  the  support  location; 
intermediate  support  consists  of  k,  and  kr . 


7.  SUMMARY  AND  CONCLUSIONS 

In  modern  high  speed  rotating  shaft  applications,  it  is  common  that  the  shaft  has  multiple 
intermediate  supports  and  discontinuities  such  as  bearings,  rotor  masses,  and  changes  in  cross- 


1323 


sections.  In  many  cases,  the  ratio  of  the  shaft  diameter  to  its  length  between  consecutive  supports 
is  large,  and  the  Timoshenko  model  (TM)  is  needed  to  accurately  account  for  the  shear  and  rotary 
inertia  effects.  In  this  paper,  the  wave  propagation  in  a  rotating,  axially  strained  Timoshenko 
shaft  model  with  multiple  discontinuities  is  examined.  The  effect  of  the  static  axial  deformation 
due  to  an  axial  load  is  also  included  in  the  model.  Based  on  results  from  Ref.  [7],  there  are  four 
possible  types  of  wave  motions  {Cases  I,  II,  III  and  IV)  in  the  Timoshenko  shaft,  as  shown  by 
Eqns.  (5a-8b).  In  practice.  Case  III  does  not  occur  and  is  excluded  in  the  analysis.  For  each 
Case,  the  wave  reflection  and  transmission  matrices  are  derived  for  a  shaft  under  various  support 
and  boundary  conditions.  Results  are  compared  with  those  obtained  by  using  the  simple  Euler- 
Bemoulli  model  (EB)  and  are  summarized  as  follows. 

1)  In  general,  the  two  shaft  models  show  good  agreement  in  the  low  frequency  range  where  the 
wave  motion  is  governed  by  Case  II  and  Case  IV.  However,  at  high  frequencies,  the  types  of 
wave  motions  and  propagation  characteristics  for  the  TM  and  EB  models  are  very  different. 

2)  The  effects  of  shaft  rotation  on  the  wave  reflection  and  transmission  are  negligible  over  the 
entire  frequency  range  and  even  at  high  speed  (up  to  44,600  rpm).  While  the  effects  of  the 
axial  load  are  significant,  especially  in  the  low  frequency  range. 

3)  When  waves  are  incident  at  supports  with  only  translational  springs,  differences  in  the  results 
between  the  TM  and  EB  models  are  small,  and  there  exists  frequency  regions  of  impedance 
matching  and  an  impedance  mismatching  frequency  (limited  to  within  the  regime  of  Case  IT). 
The  impedance  matching  and  mismatching  disappear  when  a  rotational  spring  is  added  to  the 
support.  Instead,  there  is  a  frequency  at  which  Irul  =  0  and  Ifni  =  1,  and  vibrations  on  both 
sides  of  the  support  become  strongly  coupled.  This  (delocalization)  phenomenon  suggests 
further  research  on  the  vibrations  of  constrained  multi-span  beams.  When  there  is  damping 
at  the  support,  the  frequency  at  which  IrnI  =  0  does  not  occur.  Moreover,  effects  of 
translational  damping  on  the  wave  propagation  are  more  significant  at  high  frequency, 
especially  for  the  TM  model,  however  effects  of  rotational  damping  is  not  significant  over  the 
entire  frequency  range. 

4)  Contributions  of  attenuating  wave  components  to  the  energy  in  the  reflected  and  transmitted 
waves  are  significant  when  the  shaft  is  axially  strained  and  when  the  support  has  a  rotational 
constraint.  Thus  attenuating  waves  should  be  included  in  the  formulation. 

5)  Unlike  the  spring  supports,  in  which  waves  are  easily  transmitted  at  high  frequency,  the  rotor 
mass  support  diminishes  the  wave  transmission  as  the  frequency  increases. 

6)  When  waves  are  incident  at  a  geometric  discontinuity  such  as  a  change  in  the  cross-section, 
there  are  nine  possible  combinations  of  wave  motions  on  both  sides  of  the  discontinuity.  It  is 
shown  that  differences  of  the  results  between  the  TM  and  EB  models  depend  on  the  diameter 
ratio  (and  hence  the  direction  of  the  wave  incidence).  Moreover,  incident  attenuating  waves 
contribute  more  energy  to  propagating  waves  at  the  discontinuity  when  the  waves  travel  from 
a  smaller  to  a  larger  cross-section.  When  the  shaft  is  axially  strained,  the  effects  of  the  load 
on  the  wave  propagation  are  primarily  limited  to  the  low  frequency  range. 

The  reflection  and  transmission  matrices  are  combined  with  the  transfer  matrix  method  to 
provide  a  systematic  solution  method  to  analyze  the  free  vibration  of  a  multi-span,  rotating  shaft. 
Since  the  procedure  involves  only  2x2  matrices  (while  including  the  near-field  effects  already), 
strenuous  computations  associated  with  large-order  matrices  are  eliminated. 


1324 


ACKNOWLEDGMENTS 


The  authors  wish  to  acknowledge  the  support  of  the  National  Science  Foundation  and  the 
Institute  of  Manufacturing  Research  of  Wayne  State  University  for  this  research  work. 


REFERENCES 

1.  Lin,  Y.K.,  Free  Vibrations  of  a  Continuous  Beam  on  Elastic  Supports.  International  Journal 
of  Mechanical  Sciences,  1962,  4,  pp.  409-423. 

2.  Graff,  K.F.,  Wave  Motion  in  Elastic  Solids,  Ohio  State  University  Press,  1975. 

3.  Cremer,  L,,  Heckl,  M.  and  Ungar  E.E.,  Structure-Bome  Sound,  Springer-Verlag,  Berlin, 
1973. 

4.  Fahy,  F.,  Sound  and  Structural  Vibration,  Academic  Press,  1985. 

5.  Mead,  D.J.,  Waves  and  Modes  in  Finite  Beams:  Application  of  the  Phase-Closure  Principle. 
Journal  of  Sound  and  Vibration,  1994, 171,  pp.  695-702. 

6.  Mace,  B.R.,  Wave  Reflection  and  Transmission  in  Beams.  Journal  of  Sound  and  Vibration, 
1984,  97,  pp.  237-246. 

7.  Kang,  B.  and  Tan,  C.A.,  Elastic  Wave  Motions  in  an  Axially  Strained,  Infinitely  Long 
Rotating  Timoshenko  Shaft.  Journal  of  Sound  and  Vibration  (submitted),  1997. 

8.  Han,  R.P.S.  and  Zu,  J.W.-Z.,  Modal  Analysis  of  Rotating  Shafts:  A  Body-Fixed  Axis 
Formulation  Approach.  Journal  of  Sound  and  Vibration,  1992, 156,  pp.  1-16. 

9.  Zu,  J.W.-Z.  and  Han,  R.P.S. ,  Natural  Frequencies  and  Normal  Modes  of  a  Spinning 
Timoshenko  Beam  With  General  Boundary  Conditions.  Transactions  of  the  American 
Society  of  Mechanical  Engineers,  Journal  of  Applied  Mechanics,  1992,  59,  pp.  197-204. 

10.  Katz,  R.,  Lee,  C.W.,  Ulsoy,  A.G.  and  Scott,  R.A.,  The  Dynamic  Response  of  a  Rotating 
Shaft  Subject  to  a  Moving  Load.  Journal  of  Sound  and  Vibration,  1988, 122,  pp.  131-148. 

11.  Tan,  C.A.  and  Kuang,  W.,  Vibration  of  a  Rotating  Discontinuous  Shaft  by  the  Distributed 
Transfer  Function  Method.  Journal  of  Sound  and  Vibration,  1995, 183,  pp.  451-474. 

12.  Argento,  A.  and  Scott,  R.A.,  Elastic  Wave  Propagation  in  a  Timoshenko  Beam  Spinning 
about  Its  Longitudinal  Axis.  Wave  Motion,  1995,  21,  pp.  67-74. 

13.  Dimentberg,  F.M.,  Flexural  Vibrations  of  Rotating  Shafts,  Butterworth,  London,  1961. 

14.  Dimarogonas,  A.D.  and  Paipeties,  S.A.,  Analytical  Method  in  Rotor  Dynamics,  Applied 
Science,  New  York,  1983. 

15.  Lee,  C.W.,  Vibration  Analysis  of  Rotors,  Kluwer  Academic  Publishers,  1993. 

16.  Choi,  S.H.,  Pierre,  C.  and  Ulsoy,  A.G.,  Consistent  Modeling  of  Rotating  Timoshenko  Shafts 
Subject  to  Axial  Loads.  Journal  of  Vibration  and  Acoustics,  1992, 114,  pp.  249-259. 

17.  Bhashyam,  G.R.  and  Prathap,  G.,  The  Second  Frequency  Spectrum  of  Timoshenko  Beams. 


1325 


Journal  of  Sound  and  Vibration,  1981, 76,  pp.  407-420. 

18.  Riedel,  C.H.  and  Tan,  C.  A.,  Mode  Localization  and  Delocalization  of  Constrained  Strings 
and  Beams.  Proceedings  ofASME  Biennial  Conference  on  Mechanical  Vibration  and  Noise 
(submitted),  1997 


1326 


For  simple  and  clamped  supports,  the  reflection  and  transmission  matrices  are  listed  as  follows. 


Simple  support  ( k.  =  k.  =  n 

l  =  C, 

=  c,  =  . 

Case  //  ( A  >  0 ,  B  <  0): 

1 

■  r. 

r,  ■ 

iF,  r2 

'ir, 

r./ 

'"ir.-r2 

.-ir^ 

-r^. 

CaseIV{A<0,  B<0): 

1 

-ire 

1  1 

t  “ 

'r. 

-ir: 

iT,  +r2 

.-^2 

-n_ 

i  ^T,+r2 

-r2 

•  Clamped  support  {k^  =<=<>,  m=c,  =  c,  =  J„,  =  0 );  t  =  0 . 

CaseII{A>0,  B<0): 

1  ^^1+^2  2r2 

-2iT,  -(ir,  +  r,)J’ 

CaseIV{A<0,  B<0): 

1  r-(ir,-r2)  -2iTi  ' 

^“iT.+r.L  -2r2  iT, -r^  ■ 


(32a*,  b*) 


(33a*,  b*) 


(35*) 


(36*) 


APPENDIX  II 

The  reflection  and  transmission  matrices  for  a  wave  incident  upon  a  cross-sectional  change 
for  the  simple  Euler-Bernoulli  shaft  model  can  be  determined  by  solving  the  following  sets  of 
matrix  equations.  Only  two  representative  combinations  are  shown. 

CaseII(Ai>0,  B,  <0)- Case  II  {A,>0,  B,<0): 


1327 


Casen{A,>0,  <Q)  -  Case  TV  {A,  <0 ,  <0): 


1 

1 

1 

1 1  . 

1 

1  1  . 

■  + 

- 

•^2,. 

-F^  F" 
^1/  •*■2/ 

C* 

+ 

F2  1 

rC"  = 

[T^  _r3 

L  “  1/  ^  21 

2/ J 

r,rj 

where,  F sr  and  T2r  have  been  defined  in  Eqns.  (45c,  d),  and  A^-  and  Br  are  given  by 

O’" 

G 


(39a') 

(39b') 

(46a') 

(46b*) 


If  the  rotating  shaft  is  strain-free,  then  r  can  be  reduced  to  simple  forms  representing  typical 
boundary  conditions  such  as  simple  support,  clamped  support,  and  free  end  as  shown  in  Ref.  [6]. 
Note  that  for  those  supports  in  the  strain-free  case,  the  reflection  matrices  are  constant. 


1328 


ANALYTICAL  MODELLING  OF  COUPLED  VIBRATIONS  OF 
ELASTICALLY  SUPPORTED  CHANNELS 

Yavuz  YAMAN 

Department  of  Aeronautical  Engineering,  Middle  East  Technical  University 
0653 1  Ankara,  Turkey 

An  exact  analytical  method  is  presented  for  the  analysis  of  forced  vibrations 
of  uniform  thickness,  open-section  channels  which  are  elastically  supported  at 
their  ends.  The  centroids  and  the  shear  centers  of  the  channel  cross-sections  do 
not  coincide;  hence  the  flexural  and  the  torsional  vibrations  are  coupled.  Ends 
of  the  channels  are  constrained  with  springs  which  provide  finite  transverse, 
rotational  and  torsional  stiffnesses.  During  the  analysis,  excitation  is  taken  in 
the  form  of  a  point  harmonic  force  and  the  channels  are  assumed  to  be  of  type 
Euler-Bernoulli  beam  with  St.Venant  torsion  and  torsional  warping  stiffness. 
The  study  uses  the  wave  propagation  approach  in  constructing  the  analytical 
model.  Both  uncoupled  and  double  coupling  analyses  are  performed.  Various 
response  and  mode  shape  curves  are  presented. 

1.  INTRODUCTION 

Open-section  channels  are  widely  used  in  aeronautical  structures  as  stiffeners. 
These  are  usually  made  of  beams  in  which  the  centroids  of  the  cross-section 
and  the  shear  centers  do  not  coincide.  This,  inevitably  leads  to  the  coupling  of 
possible  flexural  and  torsional  vibrations.  If  the  channels  are  symmetric  with 
respect  to  an  axis,  the  flexural  vibrations  in  one  direction  and  the  torsional 
vibrations  are  coupled.  The  flexural  vibrations  in  mutually  perpendicular 
direction  occur  independently.  In  the  context  of  this  study,  this  type  of 
coupling  is  referred  to  as  double-coupling.  If  there  is  no  cross-sectional 
symmetry,  all  the  flexural  and  torsional  vibrations  are  coupled.  This  is  called 
as  triple-coupling.  The  coupling  mechanism  alters  the  otherwise  uncoupled 
response  characteristics  of  the  structure  to  a  great  extent. 

This  problem  have  intrigued  the  scientists  for  long  time.  Gere  et  al  [1],  Lin 
[2],  Dokumaci  [3]  and  Bishop  et  al  [4]  developed  exact  analytical  models  for 
the  determination  of  coupled  vibration  characteristics.  All  those  works,  though 
pioneering  in  nature,  basically  aimed  to  determine  the  free  vibration 
characteristics  of  open-section  channels. 

The  method  proposed  by  Cremer  et  al  [5]  allowed  the  determination  of 
forced  vibration  characteristics,  provided  that  the  structure  is  uniform  in 
cross-section.  The  use  of  that  method  was  found  to  be  extremely  useful  when 
the  responses  of  uniform  structures  to  point  harmonic  forces  or  line  harmonic 
loads  were  calculated.  Mead  and  Yaman  presented  analytical  models  for  the 


1329 


analysis  of  forced  vibrations  of  Euler-Bernoulli  beams  [6].  In  that  they 
considered  finite  length  beams  ,  being  periodic  or  non-periodic,  and  studied 
the  effects  of  various  classical  or  non-classical  boundary  conditions  on  the 
flexural  response.  Yaman  in  [7]  developed  mathematical  models  for  the 
analysis  of  the  infinite  and  periodic  beams,  periodic  or  non-periodic  Kirchoff 
plates  and  three-layered,  highly  damped  sandwich  plates. 

Yaman  in  [8]  also  developed  analytical  models  for  the  coupled  vibration 
analysis  of  doubly  and  triply  coupled  channels  having  classical  end  boundary 
conditions.  In  that  the  coupled  vibration  characteristics  are  expressed  in 
terms  of  the  coupled  wave  numbers  of  the  structures.  The  structures  are  first 
assumed  to  be  infinite  in  length,  and  hence  the  displacements  due  to  external 
forcing(s)  are  formulated.  The  displacements  due  to  the  waves  reflected  from 
the  ends  of  the  finite  structure  are  also  separately  determined.  Through  the 
superposition  of  these  two,  a  displacement  field  is  proposed.  The  application 
of  the  end  boundary  conditions  gives  the  unknowns  of  the  model.  The 
analytical  method  yields  a  matrix  equation  of  unknowns  which  is  to  be  solved 
numerically.  The  order  of  the  matrix  equation  varies  depending  on  the  number 
of  coupled  waves.  If  the  cross-section  is  symmetric  with  respect  to  an  axis 
(  double-coupling)  and  if  the  warping  constraint  is  neglected,  the  order  is  six. 
If  there  is  no  cross-sectional  symmetry  (triple-coupling)  and  if  one  also 
includes  the  effects  of  warping  constraint,  the  order  then  becomes  twelve. 
This  order  is  independent  of  the  number  of  externally  applied  point  forces. 
Although  the  method  is  basically  intended  to  calculate  the  forced  response 
characteristics,  it  conveniently  allows  the  computation  of  free  vibration 
characteristics  as  well.  The  velocity  or  acceleration  of  a  point  can  easily  be 
found.  The  mode  shapes  can  also  be  determined.  Both  undamped  and  damped 
analyses  can  be  undertaken. 

This  study  is  based  on  the  models  developed  in  reference  [8]  and  aims  to 
analyze  the  effects  of  non-classical  end  boundary  conditions  on  the  coupled 
vibratory  responses.  If  the  ends  are  elastically  supported  (which  may  also  have 
inertial  properties)  the  problem  becomes  so  tedious  to  tackle  through  the 
means  of  classical  analytical  approaches.  The  current  method  alleviates  the 
difficulties  encountered  in  the  consideration  of  complex  end  boundary 
conditions. 


In  this  study  a  typical  channel,  assumed  to  be  of  type  Euler-Bernoulli  beam,  is 
analyzed.  It  represents  the  double-coupling.  Effects  of  the  elastic  end 
boundary  conditions  on  the  resonance  frequencies,  response  levels  and  mode 
shapes  are  analyzed.  Characteristics  of  otherwise  uncoupled  vibrations  are 
also  shown. 


1330 


2.  THEORY 


2.1  Flexural  Wave  Propagation  in  Uniform  Euler-Bernoulli  Beams 

Consider  a  uniform  Euler-Bernoulli  beam  of  length  L  which  is  subjected  to  a 
harmonically  varying  point  force  Fo  e  acting  at  x=Xf.  The  total  flexural 
displacement  of  the  beam  at  any  Xr  (0  <  Xr  <  L)  can  be  found  to  be  [5-8], 


w(x,,t)  =  (  i;A„e''„\  +  Fo  i  ane'^' 


r  r 


)e' 


(1) 


The  first  series  of  the  equation  represents  the  effects  of  four  waves  which  are 
being  reflected  from  the  ends  of  the  finite  beam.  They  are  called  free-waves. 
The  second  series  accounts  for  the  waves  which  are  being  created  by  the 
application  of  the  external  force  Fo  e  on  the  infinite  beam.  Those  waves  are 
known  as  forced-waves,  kn  is  the  n’th  wave  number  of  the  beam  and 
kn  =(mco^/EI)'^'‘  where  m=  Mass  per  unit  length  of  the  beam,  co=  Angular 
frequency,  EI=  Flexural  rigidity  of  the  beam,  an  values  are  the  complex 
coefficients  which  are  to  be  found  by  satisfying  the  relevant  compatibiliy  and 
continuity  conditions  at  the  point  of  application  of  the  harmonic  force  [6,7]. 
An  values,  on  the  other  hand  are  the  complex  amplitudes  of  the  free  waves  and 
are  found  by  satisfying  the  required  boundary  conditions  at  the  ends  of  the 
beam.  Once  determined,  their  substitution  to  equation  (1)  yields  the  flexural 
displacement  at  any  point  on  the  finite  beam  due  to  a  transversely  applied 
point  harmonic  force.  More  comprehensive  information  can  be  found  in  [7]. 

2.2  Torsional  Wave  Propagation  in  Uniform  Bars 

If  one  requires  to  determine  the  torsional  displacements  generated  by  a  point, 
harmonically  varying  torque,  a  similar  approach  to  the  one  given  in  Section 
2.1  can  be  used.  In  that  case,  the  total  torsional  displacement  can  be  written  as: 


=  )e‘”'  (2) 

k  is  the  wave  number  of  the  purely  torsional  wave  and  is  known  to  be 
k=(-pIoa)VGJ)^^^  .  k2=  -ki  and  GJ=Torsional  rigidity  of  the  beam,  p=Material 
density,  Io=Polar  second  moment  of  area  of  the  cross-section  with  respect  to 
the  shear  centre.  Toe'“  ^  is  the  external  harmonic  torque  applied  at  x=Xt  and 
b=l/(2kGJ).  Bn  values  are  the  complex  amplitudes  of  the  torsional  free-waves 
and  are  found  by  satisfying  the  appropriate  end  torsional  boundary  conditions. 

The  consideration  of  the  warping  constraint  To  modifies  equation  (2)  to  the 
following  form. 


1331 


(3) 


«x„t)  =  (£C„e^\  +  Toic„e-'=„'V,'  )  e 

n*l  »•! 

Now  kn  are  the  roots  of 


EFo  kn'^-GJkn^-pIoCO^=0 


(4) 


Cn  values  are  found  by  satisfying  the  necessary  equilibrium  and  compatibility 
conditions  at  the  point  of  application  of  the  point  harmonic  torque  acting  on  an 
infinite  bar  [8].  Cn  values  are  determined  from  the  end  torsional  boundary 
conditions  of  the  finite  bar. 

2.3  End  Boundary  Conditions  for  Uncoupled  Vibrations 


2.3.1  Purely  Flexural  Vibrations 

Consider  an  Euler-Bernoulli  beam  of  length  L  which  is  supported  by  springs  at 
its  ends.  The  springs  provide  finite  transverse  and  rotational  constraints  Kt  and 
Kr  respectively.  The  elastic  end  boundary  conditions  can  be  foimd  to  be: 

El  w”(0)  -  Kr  I  w’(0)=0  El  w’”(0)  +  Ktj  w(0)=0 

El  w”(L)  +  K  r,r  w’(L)=0  El  w’”(L)  -  Kt,rW(L)=0  (5) 


Here  w’= 


dw(x) 


w 


d‘w(x) 


and  w’” 


dV(x) 


w(x)  is  the  spatially 


dx  ’  dx“  dx' 

dependent  part  of  equation  (1)  and  second  subscripts  1  and  r  allows  one  to  use 
different  stiffnesses  for  left  and  right  ends.  A  more  comprehensive  study  on 
these  aspects  can  be  found  in  references  [6,7]. 


2.3.2  Purely  Torsional  Vibrations 

Now  consider  a  bar  of  length  L  which  is  supported  by  torsional  springs, 
having  finite  Ktor,at  its  ends.  The  elastic  end  boundary  conditions  requires  that, 

Torque  (0)  -  Ktor,i  (|)(0)=0  and  Torque  (L)  +  Ktor  .r  (i)(L)=0  (6) 

Depending  on  the  consideration  of  the  warping  constraint  To,  the  torque  has 
the  following  forms 

Torque(x)=GJ  — —  or  Torque(x)=  GJ  — —  -  Ei  o  ,  3  ( /) 

dx  dx  ux 


1332 


(})(x)  in  equation  (7)  should  be  obtained  either  from  equation  (2)  or  equation  (3) 
depending  on  the  warping  constraint  r©. 

2.4  Doubly-Coupled  Vibrations 

Now,  consider  Figure  1  .  It  defines  a  typical  open  cross-section  which  is 
synunetric  with  respect  to  y  axis 


V  z 


(b) 

Figure  1  :  A  Typical  Cross-section  of  Double-coupling 

(  a.  Coordinate  System,  b.  Real  and  Effective  Loadings 
C:  Centroid,  O:  Shear  Centre) 

A  transverse  load  applied  through  C  results  in  a  transverse  load  through  O  and 
a  twisting  torque  about  O.  In  this  case  the  flexural  vibrations  in  z  direction  are 
coupled  with  the  torsional  vibrations  whereas  the  flexural  vibrations  in  y 
direction  occur  independently.  The  motion  equation  of  the  coupled  vibrations 
is  known  to  be  [1,2]. 


a-w  a-(j) 


=  0 


a‘‘(i)  3^6  a’w  a"(j) 


(8) 


If  one  assumes  that, 


w(x,t)  =  w„eVe”' 

(|)(x.t)  =  4>„eVe“’  (9) 


1333 


Then,  it  can  be  found  that,  a  load  Pz  through  the  centroid  will  create  the 
following  displacements  at  any  x  (0  <  x  <  L)  along  the  length  of  the 
channel  [8], 

w(x,t)  =  (2A„e^’‘+Pjt  a„  e'^  )e'"' 

n^l  nvl 

n«l  n>I 

Now  kn  values  are  the  coupled  wave  numbers,  An  values  are  the  complex 
amplitudes  of  the  coupled  free  waves,  an  values  are  the  complex  coefficients 
which  are  to  be  found  by  satisfying  the  required  compatibility  and  continuity 
conditions  and  'Pn^  (  (El^  kn'^-mco^)  /  (CymcD^) )  [8]. 

If  required,  the  warping  displacement  u(x,t)  can  be  found  from  (l)(x,t  )as 

u(x,t)=-2A,^iM  (11) 

dx 

where  As  is  the  swept  area. 

Here  2j  gives  the  order  of  the  motion  equation.  j=3  defines  the  case  in  which 
the  effects  of  warping  constraint  are  neglected  and  j=4  represents  the  case 
which  includes  the  warping  effects. 


An  values  are  found  by  satisfying  the  necessary  2j  end  boundary  conditions.  If 
warping  constraint  is  neglected,  the  required  six  boundary  conditions  have  the 
general  forms  given  in  equations  (5)  and  (6).  But  the  forms  of  w(x)  and  (j)(x) 
are  now  those  given  by  equations  (10)  with  j=3.  If  the  warping  constraint  is 
included  in  the  analysis  the  boundary  conditions  become  eight.  The  six  of 
those  are  again  found  by  considering  equations  (10)  with  j=4  and  substituting 
the  resultant  forms  into  equations  (5)  and  (6).  The  remaining  two  can  be 
found  by  evaluating  equation  (1 1)  at  both  ends. 

When  the  flexural  and  torsional  displacement  expressions  are  substituted  into 
the  relevant  equations,  a  set  of  equations  is  obtained.  For  the  case  of  a  load 
Pz  and  no  warping  constraint,  the  following  equations  can  be  found  for  j=3. 

EI^  w’”(0)  +  Kt,iw(0)=0 

EI^  (E  kn^  An+  (-1)  PzZ  -  kn^  a  „  6  '  "f  '  ) 

^t,l  (  E  ■^n  ■^PzE^n^  n  f  )  “0  (12) 


1334 


(13) 


EI^w”(0)  -Kr,i  w’(0)=0: 

EI^(  I  kn'An  +  Pz  t  kn'an  6 ‘  '  ) 

n«l  n«t 

■  (  S  kn  An  +  Pz^'kn^n®  n  f  ) 

n*l  n“l 


GJ^^l«=o-K,„,.i  W)=0: 

dx 

GJ(  S  k„  T „  A„  +  (- 1 )  (Pz  cy)  t  -k„  'f  „  a„  e  '  ’‘f ') 

fl«l  11' I 

-K,or,i  (  i  %  A„  +  (Pz  Cy)  i  >?„  a„  e  ')=0 


diKx)  I 

GJ-^Ix.l  +  K,„„  <KL)=0: 
dx 

GJ(  2k„>P„  A„  e“+  (PzCy)  i-k„'P„a„  ) 

ns  I  11=1 

+  K,„,.r  ( I  %  A„  +  (Pz  Cy)  t  Tn  a„  e  "f  ')=0 


El5W”(L)+  Kr,rW’(L)=0: 

El^CS  k„^A„e^‘-+(Pz)  i  k„^a„  ) 

n-l  ii’*) 

+  Kz,r(  i  k„A„  e“+  (Pz)  t-k„a„  e■^'‘-^')=0 

n»l  n=i 


(14) 


(15) 


(16) 


EI^  w’”(L)-Kt,rW(L)=0: 

Eiaz  kn'Ane'n^  +(Pz)  2  -  kn^  a  „  e n  ' ^  ) 

11=1  Iia] 

-K,|(Z  A„eV+(Pz)  Za„e^''-V)=0  (17) 

lt=l  11=1 

Here  (-1)  multipliers  are  included  due  to  the  symmetry  and  anti-symmetry 
effects. 

Those  equations  can  be  cast  into  the  following  matrix  form. 


=  -{Terms  containing  Pz}  (18) 


1335 


An  eighth  order  equation  represents  the  necessary  matrix  equation  for  the 
determination  of  An  values  if  the  warping  constraint  is  included  in  the  analysis. 
In  that  case,  equations  (12),  (13),  (16)  and  (17)  are  valid  with  j=4.  On  the  other 
hand  equations  (14)  and  (15)  should  be  replaced  by, 

(GJ^-Er„^^)L.o  <l>(0)=0: 

dx  dx 

(GJ(  £  k„  %  A„  +  (- 1 )  (P.  Cy)  £  -k„  a„  e  *  „ '  ’‘f ')  - 

nol  n»l 

Er„(  £  k„3  'f  „  A„  +  (- 1 )  (P^  Cy)  £  -k„  =  %  a„  e  *  „ '  -f ')) 

it«l  IIS’! 

-K,„,,,(|;'P„A„+(P,Cy)i  'P„a„e-^'’‘f')=0  (19) 

cr  A/T^=A. 

(GJ  “  EFo  j  3  )•  x=L  I^tor>r  y(L)  0  . 
dx  dx 

(GJ(  Jkn'Fn  A„  e’=„‘-+  (PjCy)  i-kn'Pnan  '  )- 

n=t  Ii»l 

EFoC  Xkn^  'J'n  A„  eV+  (PzCy)  £-k„^'P„a„  ' )) 

11=11  n-l 

+  Kior,,  (  X  %  A„  +  (Pz  Cy)  X  %  a„  e  *  „ '  \  ')=0  (20) 

n-]  ii3| 

where  j=4.  The  remaining  two  equations  are  found  by  considering  the  warping 
of  the  extreme  ends.  If  the  ends  are  free  to  warp  the  axial  stress  is  zero,  if  the 
ends  are  not  to  warp  the  axial  displacements  are  zero  at  both  ends.  No  elastic 
constraints  are  imposed  on  end  warping.  If  the  left  end  is  free  to  warp  and  the 
right  end  is  not  to  warp,  the  required  boundary  conditions  can  be  shown  to  be; 

u’(0)=0:(x  kn^'Pn  A„  +  PzX  'i'.ian  k„^e-^  '  ^')  =  0  (21) 

u(L)=0  :(X  k„  e^‘'  %  A„  +  PzX  (-k„)  '‘-"f')  =  0  (22) 

n=]  list 

All  the  equations  can  be  put  into  the  following  matrix  form 

A.' 

A: 

A, 

•  =  -{Terms  containing  Pz}  (23) 

A, 

A. 

A. 

A. 


1336 


Required  An  values  are  numerically  found  from  equations  (18)  or  (23).  Their 
substitution  to  the  appropriate  forms  of  equations  (10)  and  (11)  yield  the 
required  responses  at  any  point  on  the  beam. 

3.RESULTS  AND  DISCUSSION 

The  theoretical  model  used  in  the  study  is  shown  in  Figure  1  and  has  the 
following  geometric  and  material  properties: 

L=l(m),  A  =1.0*10"' (m\  h  =  5.0*10-^(m),  15  =  4.17*10-*  (m\ 

Cy  =  15.625*10'*  (m),  J  =  3.33* lO"' '  (m*),  =7.26*10'*  (m"'),  p=  2700  (kg/m*), 
r„  =2.85*10''*  (m*),  E  =  7*10'“  (N/m*),  G  =  2.6*  10'“  (N/m*). 

Structural  damping  for  torsional  vibrations  is  included  through,  complex 
torsional  rigidity  as  GJ*=GJ(l+z|3).  For  coupled  vibrations,  it  is  also  included 
through  the  complex  flexural  rigidity  as  El^(H-z'n). 

First  presented  are  the  results  for  purely  torsional  vibrations.  A  bar  assumed  to 
have  the  given  L,  p,  lo,  G  and  J  values  is  considered.  The  bar  is  then  restrained 
at  both  ends  by  springs  having  the  same  torsional  stiffness  Ktor-  A  very  low 
damping,  p=10'^,  is  assigned  and  the  resonance  frequencies  are  precisely 
determined.  It  is  found  that,  the  introduction  of  a  small  Ktor  introduces  a  very 
low  valued  resonance  frequency.  That  fundamental  frequency  increases  with 
increasing  Ktor  and  as  torsional  constraint  reaches  to  very  high  values,  it 
approaches  to  the  fundamental  natural  frequency  of  torsionally  fixed-fixed 
beam.  Table  1  gives  the  fundamental  frequencies  for  a  range  of  Ktor  values  and 
Figure  2  represents  the  fundamental  mode  shapes  for  selected  Ktor  values. 


Table  1:  Uncoupled  Fundamental  Torsional  Resonance  Frequencies 

((3=10’^,No  Warping  Constraint) 


Ktor  [N1 


Frequency  [Hz] 


0 

0. 

10'^ 

1.606 

10'' 

5.035 

io“ 

14.683 

10' 

28.407 

10^ 

32.678 

10^ 

33.187 

10^ 

33.239 

;10*“ 

33.245 

1337 


Figure  6  on  the  other  hand  represents  the  low  frequency  torsional  receptances 
of  the  case  in  which  the  warping  constraint  is  taken  into  consideration  and 
the  ends  are  free  to  warp.  This  graph  is  included  in  order  to  show  the  variation 
of  fundamental  torsional  resonance  frequencies  for  a  range  of  Ktor  values. 


_ -  K,3r= 

K„,= 

kK,„= 


5*10': 

1*10': 

5-10" 

1-10' 

5-10' 

1*10® 


Figure  6.  Fundamental  Frequencies  of  Purely  Torsional  Vibrations 
((3=0.01,  x=0.13579[m], Warping  Constraint  Included,  Ends  are  Free  to  Warp) 


The  second  part  of  the  study  investigates  the  characteristics  of  doubly-coupled 
vibrations.  Now,  the  effects  of  each  constraint  are  separately  considered.  A 
channel  having  the  given  parameters  is  supported  at  its  ends  by  springs  Kt,  Kr, 
and  Ktor-  Warping  constraint  is  included  in  the  analysis  and  the  ends  are 
assumed  to  warp  freely.  First  analyzed  is  the  effects  of  Ktor-  For  this  Kt=10^° 
[N/m]  and  Kr  =10^°  [N]  are  assigned  at  both  ends  of  the  channel  and  kept  fixed 
throughout  the  study.  Ktor  is  varied  and  the  frequencies  are  shown  in  Table  2. 


Table  2.  Effects  of  Ktnr  in 

Doublv-counled  Vibrations 

(Kt=10^°  [N/m]  and  Kr=10^®  [N],  Warping  Constraint  Included) 

A:  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] 

Ktor  [Nl 

A 

B 

10-^ 

1.607 

134.603 

10'' 

5.069 

134.936 

10° 

15.620 

138.258 

10' 

39.725 

169.247 

10^ 

56.691 

318.423 

10^ 

59.528 

422.061 

10^ 

59.828 

430.642 

1338 


0.0  0.2  0.4  0.6  0.8  1.0 


ND  LENGTH 

Figure  2.  Fundamental  Mode  Shapes  of  Purely  Torsional  Vibrations 
((3=0,  No  Warping  Constraint) 

Then,  the  warping  constraint  To  is  included  in  the  analysis  and  the  results  of 
purely  torsional  vibrations  are  presented  again.  The  beam  had  the  same  Ktor 
values  at  both  ends  and  the  numerical  values  of  the  relevant  parameters  are 
taken  to  be  those  previously  defined.  Figure  3  represents  the  fundamental 
mode  shapes  for  which  the  ends  are  free  to  warp,  whereas  Figure  4  shows  the 
mode  shapes  of  the  case  in  which  there  is  no  warping  at  the  ends. 


Figure  3.  Fundamental  Mode  Shapes  of  Purely  Torsional  Vibrations 


(P=0,  Warping  Constraint  Included,  Ends  are  Free  to  Warp) 


(P=0,  Warping  Constraint  Included,  Ends  Can  Not  Warp) 


Figure  5  is  drawn  to  highlight  the  effects  of  end  warping.  Both  ends  of  the 
channel  are  restrained  with  Ktor=l  *  lO’  [N]  and  all  the  other  parameters  of  the 
study  are  kept  fixed.  Figure  5  represents  the  direct  torsional  receptances  of 
two  cases  in  which  the  ends  of  the  channel  are  allowed  to  warp  and  not  to 
warp  in  turn.  It  can  be  seen  that  the  prevention  of  end  warping  increases  the 
resonant  frequencies. 


Figure  5.  Frequency  Response  of  Purely  Torsional  Vibrations 

([3=0.01,  X  =0.13579  [m],  Ktor=l*10'  [N]»  Warping  Constraint  Included) 


1340 


It  can  be  seen  that,  when  it  has  lower  values  Ktor  is  more  effective  on  the 
torsion  dominated  resonance  frequencies.  For  the  higher  Ktor  values,  the 
effects  are  more  apparent  on  the  flexure  dominated  frequencies. 

Figure  7  represents  the  direct  flexural  receptance  of  the  channel  for 
a  set  of  selected  end  stiffnesses.  Torsion  dominated  resonances  at  59.528  [Hz], 
206.071  [Hz]  and  476.649  [Hz]  appear  as  spikes.  The  flexure  dominated 
resonance  occurs  at  422.061  Hz. 


Figure  7.  Frequency  Response  of  Doubly-coupled  Vibrations 
(ti=0.001,  P=0.001,  x=0.13579[m],  Warping  Constraint  Included, 

Ends  are  free  to  warp,  Kr=1.10^°[N],  Kt=1.10^°  [N/m],  Ktor=l-10^  [N] ) 

Then  the  effects  of  the  rotational  spring,  Kr,  are  considered.  The  ends  of  the 
channel  are  assumed  to  be  restrained  with  Kt=10^°  [N/m]  and  Ktor  =10^°  [N]. 
The  resulting  frequencies  are  given  in  Table  3  for  a  range  of  Kr  values. 


Table  3.  Effects  of  Kr  in  Doublv-coupled  Vibrations 

(Kt=10^°  [N/m]  and  Ktor=10^°  [N],  Warping  Constraint  Included) 
A;  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] _ 


Kr[N] _ A _  B 


10*^ 

58.678 

205.344 

10° 

58.679 

205.358 

10^ 

58.699 

206.688 

10^ 

58.858 

218.055 

10" 

59.437 

289.738 

10^ 

59.799 

405.667 

10^ 

59.855 

429.345 

1341 


It  is  seen  that  Kr  is  not  effective  on  torsion  dominated  resonance  frequencies, 
but  plays  significant  role  for  flexure  dominated  resonance  frequencies. 

Finally  considered  the  effects  of  the  transverse  spring  Kt.  Again,  the  channel  is 
assumed  to  have  very  high  Kr  and  Ktor  values  at  both  ends  and  Kt  values  are 
varied.  Table  4  shows  the  resonance  frequencies. 


Table  4.  Effects  of  Kt  in  Doublv-coupled  Vibrations 


(Kr=10^°  [N]  and  Ktor =10^°  [N],  Warping  Constraint  Included) 
A:  First  Torsion  Dominated  Frequency  [Hz] 

B:  First  Flexure  Dominated  Frequency  [Hz] _ 


Kt  [N/m] 

A 

B 

10' 

1.369 

70.455 

10^ 

4.328 

70.501 

10^ 

13.592 

70.973 

10^ 

39.282 

77.379 

10= 

58.095 

159.665 

10= 

59.699 

361.255 

10* 

59.860 

431.215 

It  can  be  seen  that  the  transverse  stiffness,  like  torsional  stiffness,  effects  both 
flexure  and  torsion  dominated  frequencies. 


4.CONCLUSIONS 


In  this  study,  a  new  analytical  method  is  presented  for  the  analysis  of  forced 
vibrations  of  open  section  channels  in  which  the  flexible  supports  provide  the 
end  constraints.  The  dynamic  response  of  open  section  channels  is  a  coupled 
problem  and  their  analysis  requires  the  simultaneous  consideration  of  all  the 
possible  vibratory  motions.  The  wave  propagation  approach  is  an  efficient  tool 
for  this  complicated  problem  and  the  developed  method  is  based  on  that. 

The  current  method  analyzes  the  forced,  coupled  vibrations  of  open  section 
channels.  The  channels,  taken  as  Euler-Bernoulli  beams,  have  uniform  cross- 
section  and  a  single  symmetry  axis.  That  consecutively  leads  to  the  coupling 
of  flexural  vibrations  in  one  direction  and  torsional  vibrations.  The  excitation 
is  assumed  to  be  in  the  form  of  a  harmonic  point  force,  acting  at  the  centroid. 


1342 


Various  frequency  response  curves  of  uncoupled  and  coupled  vibrations  are 
presented  for  a  variety  of  different  elastic  end  boundary  conditions(  which  may 
also  have  the  inertial  properties).  The  developed  method,  although  aimed  at 
determining  the  forced  vibration  characteristics,  is  also  capable  of  determining 
the  free  vibration  properties.  This  is  also  demonstrated  by  presenting  various 
mode  shape  graphs.  It  has  been  determined  that  the  transverse  and  the  torsional 
stiffnesses  play  more  significant  role  as  compared  to  the  rotational  stifness. 

The  method  can  be  used  in  analyzing  the  effects  of  multi  point  and/or 
distributed  loadings.  This  can  simply  be  achieved  by  modifying  the  terms  of 
the  forcing  vector  without  increasing  the  order  of  the  relevant  matrix  equation. 
The  developed  method  can  also  be  used  in  the  analysis  of  elastically 
supported,  triply-coupled  vibrations  of  uniform  channels.  Results  of  that  study 
will  be  the  subject  of  another  paper. 


REFERENCES 

1.  Gere,  J.M.  and  Lin,  Y.K.,  Coupled  Vibrations  of  Thin-Walled  Beams  of 
Open  Cross-Section.  J.  Applied Mech  Trans. ASME.,\9SZ,  80,373-8. 

2.  Lin,  Y.K.,  Coupled  Vibrations  of  Restrained  Thin-Walled  Beams. 

J.  Applied  Mech.  Trans. ASME.,  1960,  82,  739-40. 

3.  Dokumaci,  E.,  An  Exact  Solution  for  Coupled  Bending  and  Torsional 
Vibrations  of  Uniform  Beams  Having  Single  Cross-Sectional  Symmetry. 
JSoundandVib.Am,  119,443-9. 

4.  Bishop,  R.E.D,  Cannon,  S.M.  and  Miao,  S.,  On  Coupled  Bending  and 
Torsional  Vibration  of  Uniform  Beams.  J.Sound  and  Fi'/)., 1989,131,457-64. 

5.  Cremer,  L.  and  Heckl,  y\..,Structure~  Borne  Sound,  Springer-Verlag,1988. 

6.  Mead,  D.J.  and  Yaman,  Y.,  The  Harmonic  Response  of  Uniform  Beams  on 
Multiple  Linear  Supports:  A  Flexural  Wave  Analysis.  J.  Sound  and  Vib, 
1990,  141,465-84 

7.  Yaman,  Y.  Wave  Receptance  Analysis  of  Vibrating  Beams  and  Stiffened 
Plates.  PA  Z).  Ttew,  University  of  Southampton,  1989. 

8  Yaman,  Y.,  Vibrations  of  Open-Section  Channels:  A  Coupled  Flexural  and 
Torsional  Wave  Analysis.  (J.  Sound  and  Vib,  Accepted  for  publication) 


1343 


1344 


THE  RESPONSE  OF  TWO-DIMENSIONAL  PERIODIC  STRUCTURES 
TO  HARMONIC  AND  IMPULSIVE  POINT  LOADING 


R.S.  Langley 

Department  of  Aeronautics  and  Astronautics 
University  of  Southampton 
Southampton  S017  IBJ 


ABSTRACT 

Much  previous  work  has  appeared  on  the  response  of  a  two-dimensional 
periodic  structure  to  distributed  loading,  such  as  that  arising  from  a  harmonic 
pressure  wave.  In  contrast  the  present  work  is  concerned  with  the  response 
of  a  periodic  structure  to  localised  forcing,  and  specifically  the  response  of  the 
system  to  both  harmonic  and  impulsive  point  loading  is  considered  by 
employing  the  method  of  stationary  phase.  It  is  shown  that  the  response  can 
display  a  complex  spatial  pattern  which  could  potentially  be  exploited  to 
reduce  the  level  of  vibration  transmitted  to  sensitive  equipment. 

1.  INTRODUCTION 

Many  types  of  engineering  structure  are  of  a  repetitive  or  periodic 
construction,  in  the  sense  that  the  basic  design  consists  of  a  structural  unit 
which  is  repeated  in  a  regular  pattern,  at  least  over  certain  regions  of  the 
structure.  An  orthogonally  stiffened  plate  or  shell  forms  one  example  of  an 
ideal  two-dimensional  periodic  structure  in  which  the  fundamental  structural 
unit  is  an  edge  stiffened  panel.  Although  a  completely  periodic  structure  is 
unlikely  to  occur  in  practice,  much  can  be  ascertained  regarding  the  structural 
dynamic  properties  of  a  real  structure  by  considering  the  behaviour  of  a 
suitable  periodic  idealization.  For  this  reason,  much  previous  work  has  been 
performed  on  the  dynamic  behaviour  of  two-dimensional  periodic  structures, 
with  particular  emphasis  on  free  vibration  and  the  response  to  pressure  wave 
excitation  [1,2].  However,  no  results  have  yet  been  appeared  regarding  the 
response  of  two-dimensional  periodic  structures  to  point  loading  (as  might 
arise  from  equipment  mounts),  and  this  topic  forms  the  subject  of  the  present 
work.  A  general  method  of  computing  the  response  to  both  harmonic  and 
impulsive  loading  is  presented,  and  this  is  then  applied  to  an  example  system. 

Initially  the  response  of  a  two-dimensional  periodic  structure  to  harmonic 
point  loading  is  considered,  and  it  is  shown  that  the  far-field  response  can  be 
expressed  very  simply  in  terms  of  the  "phase  constant"  surfaces  which 
describe  the  propagation  of  plane  waves.  It  is  further  shown  that  for 


1345 


excitation  within  a  pass  band  two  distinct  forms  of  response  can  occur;  in  the 
first  case  the  amplitude  of  the  response  has  a  fairly  smooth  spatial  distribution, 
whereas  in  the  second  case  a  very  uneven  distribution  is  obtained  and  "shadow 
zones"  of  very  low  response  are  obtained.  The  second  form  of  behaviour  is 
related  to  the  occurrence  of  caustics  (defined  in  section  3.3),  and  the 
distinctive  nature  of  the  response  suggests  that  a  periodic  structure  might  be 
designed  to  act  as  a  spatial  filter  to  isolate  sensitive  equipment  from  an 
excitation  source. 

Attention  is  then  turned  to  the  impulse  response  of  a  two-dimensional  periodic 
structure.  It  is  again  shown  that  the  response  can  be  expressed  in  terms  of  the 
phase  constant  surfaces  which  describe  the  propagation  of  plane  waves.  The 
application  of  the  method  of  stationary  phase  to  this  problem  has  a  number  of 
interesting  features,  the  most  notable  being  the  fact  that  four  or  more 
stationary  points  can  arise.  It  is  found  that  a  surface  plot  of  the  maximum 
response  amplitude  against  spatial  position  reveals  features  which  resemble  the 
"caustic"  distributions  obtained  under  harmonic  loading. 

2.  RESPONSE  TO  A  HARMONIC  POINT  LOAD 


2.1  Modal  Formulation  and  Extension  to  the  Infinite  System 
A  two-dimensional  periodic  structure  consists  of  a  basic  unit  which  is  repeated 
in  two  directions  to  form  a  regular  pattern,  as  shown  schematically  in  Figure 
1.  Each  unit  shown  in  this  figure  might  represent  for  example  an  edge 
stiffened  curved  panel  in  an  aircraft  fuselage  structure,  a  three-dimensional 
beam  assembly  in  a  roof  truss  structure,  or  a  pair  of  strings  in  the  form  of  a 
"-f"  in  a  cable  net  structure.  The  displacement  w  of  the  system  can  be 
written  in  the  form  w{n,x),  where  n={n^  nf)  identifies  a  particular  unit  and 
x  =  {Xi  X2  X3)  identifies  a  particular  point  within  the  unit.  The  coordinate 
system  x  is  taken  to  be  local  to  each  unit,  and  the  precise  dimension  of  both 
X  and  the  response  vector  w  will  depend  on  the  details  of  the  system  under 
consideration. 

The  present  section  is  concerned  with  the  response  of  a  two- 
dimensional  periodic  structure  to  harmonic  point  loading  of  frequency  co.  In 
the  case  of  a  system  of  finite  dimension,  the  response  at  location  (n,x)  to  a 
harmonic  force  F  applied  at  (0,jCo)  can  be  expressed  in  the  standard  form  [3] 


»’(n,*)=EE 

P  9 


(1) 


where  rj  is  the  loss  factor,  4>p^(n,x)  are  the  modes  of  vibration  of  the  system 
and  (j}pg  are  the  associated  natural  frequencies.  The  modes  ^p^  which  appear 
in  equation  (1)  are  scaled  to  unit  generalized  mass,  so  that 


1346 


/!,  «!  V 


(2) 


where  V  represents  the  volume  (or  equivalent)  of  a  unit  and  p(jc)  is  the  mass 
density.  The  present  concern  is  with  the  response  of  an  infinite  system,  or 
equivalently  the  response  of  a  large  finite  system  in  which  the  vibration  decays 
to  a  negligible  level  before  reaching  the  system  boundaries.  In  this  case  the 
response  is  independent  of  the  system  boundary  conditions,  and  it  follows  that 
any  analytically  convenient  set  of  modes  can  be  employed  in  equation  (1).  As 
explained  in  reference  [4],  it  is  expedient  to  consider  the  Born- Von  Karman 
(or  "periodic”)  boundary  conditions,  as  in  this  case  the  modes  of  vibration  can 
be  expressed  very  simply  in  terms  of  propagating  plane  wave  components. 
In  this  regard  it  can  be  noted  from  periodic  structure  theory  [5]  that  a 
propagating  plane  wave  of  frequency  w  has  the  general  form 

w(rt,jc)=i?^{^(jc)exp(zej«,  +ie^n^nu)t)} ,  (3) 


where  and  eo  are  known  as  the  propagation  constants  of  the  wave  (with  - 
T<e,<T  and  -7r<G2<T  for  uniqueness),  and  g{x)  is  a  complex  amplitude 
function.  By  considering  the  dynamics  of  a  single  unit  of  the  system  and 
applying  Bloch’s  Theorem  [5],  it  is  possible  to  derive  a  dispersion  equation 
which  must  be  satisfied  by  the  triad  (w, 61,62)  -  by  specifying  Gj  and  €2  this 
equation  can  be  solved  to  yield  the  admissible  propagation  frequencies  w.  By 
way  of  example,  solutions  yielded  by  this  procedure  for  a  plate  which  rests  on 
a  grillage  of  simple  supports  are  shown  in  Figure  2  (after  reference  [6]).  It 
is  clear  that  the  solutions  form  surfaces  above  the  61-62  plane  -  these  surfaces 
are  usually  referred  to  as  "phase  constant"  surfaces,  and  a  single  surface  will 
be  represented  here  by  the  equation  a;= 0(61,62).  The  phase  constant  surfaces 
always  have  cyclic  symmetry  of  order  two,  so  that  0(ei,62)=0(-6i,-62);  for  an 
orthotropic  system  the  surfaces  also  have  cyclic  symmetry  of  order  four,  and 
therefore  only  the  first  quadrant  of  the  61-62  plane  need  be  considered 
explicitly,  as  in  Figure  2. 

The  key  point  about  the  Born- Von  Karman  boundary  conditions  is  that 
a  single  propagating  wave  can  fully  satisfy  these  conditions  providing  and 
6o  are  chosen  appropriately.  The  conditions  state  that  the  left  hand  edge  of  the 
system  is  contiguous  with  the  right  hand  edge,  and  similarly  the  top  edge  is 
contiguous  with  the  bottom  edge,  so  that  the  system  behaves  as  if  it  were 
topologically  equivalent  to  a  torus.  If  the  system  is  comprised  of  XN2 
units,  then  a  propagating  wave  will  satisfy  these  conditions  if  and 

62^2 =2x^  for  any  integers  p  and  q.  Following  equation  (3),  the  displacement 
associated  with  such  a  wave  can  be  written  in  the  form 


1347 


(4) 


where  ei^  and  €2,  are  the  appropriate  values  of  the  phase  constants,  and 
<^^^=0(ej^,e2g)-  Now  since  it  follows  that  a  wave  of 

frequency  travelling  in  the  opposite  direction  to  will  also  satisfy  the 
boundary  conditions.  This  wave  say)  will  have  the  form 

where  it  has  been  noted  from  periodic  structure  theory  that  reversing  the 
direction  of  a  wave  leads  to  the  conjugate  of  the  complex  amplitude  function 
^(x).  The  two  waves  represented  by  equations  (4)  and  (5)  can  be  combined 
with  the  appropriate  phase  to  produce  two  modes  of  vibration  of  the  system 
in  the  form 


KSn,x) 

• 

Re 

Im 

(6) 


By  adopting  this  set  of  modes  it  can  be  shown  [4]  that  equation  (1)  can  be  re¬ 
expressed  as 

K.(«,x)=  £  £  2g;WF^g„(xJexp(-ie,^«,-%,n,) 

^=1-^/2  9=1  -Nj/2  +Z?7)  -(J? 


where  and  M  have  been  taken  to  be  even,  and  the  amplitude  function  gp^ 
is  scaled  so  that 

g^,=[2p(x)WV,Ar,]-%(x),  {llV)\^f„(x)f;,(x)dx=\,  (8,9) 

where  the  normalized  amplitude  function  fp^  is  defined  accordingly.  The 
summation  which  appears  in  equation  (7)  includes  only  those  modes  associated 
with  a  single  phase  constant  surface  0(€i,62);  if  more  than  one  surface  occurs 
then  the  equation  should  be  summed  over  the  complete  set  of  surfaces.  The 
summation  will  include  modes  for  each  surface,  which  is  consistent  with 
known  results  for  the  modal  density  of  a  two-dimensional  periodic  structure. 

Equation  (7)  yields  the  response  of  a  finite  system  of  dimension  XM 
to  a  harmonic  point  load  -  this  response  is  identical  to  that  of  an  infinite 
system  if  the  vibration  decays  to  a  negligible  amount  before  meeting  the 
system  boundaries.  If  the  system  size  is  allowed  to  tend  to  infinity  in  equation 


1348 


(7)  then  neighbouring  values  of  the  phase  constants  e^p  and  €2^  become  closely 
spaced  (since  deip=ei_p+i-eip=2T/A^i  and  de2g= €2,9+ 2x77/2) >  and  in  this 
case  the  summations  can  be  replaced  by  integrals  over  the  phase  constants  to 
yield 


-  LI  me„e,)ninv)-c^^ 


where  0)= 0(61,62)  and  ^(x)  is  the  complex  amplitude  associated  with  the  wave 
(£0,61,62).  The  evaluation  of  the  integrals  which  appear  in  equation  (10)  is 
discussed  in  the  following  sub-sections. 


2.2  Integration  over  61 

The  integral  over  6i  which  appears  in  equation  (10)  can  be  evaluated  by  using 
contour  integration  techniques.  Two  possible  contours  in  the  complex  e,  plane 
are  shown  in  Figure  3;  to  ensure  a  zero  contribution  from  the  segment 
Im(6i)  =  ±oo,  the  upper  contour  is  appropriate  for  /Zi<0  while  the  lower 
contour  should  be  used  for  n^X).  For  each  contour  the  contributions  from 
the  segments  and  61;;= x  cancel,  since  the  integrand  which  appears  in 

equation  (10)  is  unchanged  by  an  increment  of  2x  in  the  real  part  of  ei.  The 
only  non-zero  contribution  to  the  integral  around  either  contour  therefore 
arises  from  the  segment  which  lies  along  the  real  axis.  The  poles  of  the 
integrand  occur  at  the  61  solutions  of  the  equation 

[fl(6i,62)?(U/i7)-a;^=0,  (11) 

for  specified  62  and  oj.  By  definition  there  will  be  two  real  solutions^  in  the 
absence  of  damping  (77=0)  providing  the  frequency  range  covered  by  the 
phase  constant  surface  includes  oj.  Any  complex  solutions  to  equation  (11)  in 
the  absence  of  damping  will  correspond  physically  to  "evanescent"  waves 
which  decay  rapidly  away  from  the  applied  load.  The  present  analysis  is 
concerned  primarily  with  the  response  of  the  system  in  the  far  field  (that  is, 
at  points  remote  from  the  excitation  source),  and  for  this  reason  attention  is 
focused  solely  on  those  roots  to  equation  (11)  which  are  real  when  77=0.  The 
effect  of  damping  on  these  roots  can  readily  be  deduced:  if  77  is  small  then  it 
follows  from  equation  (11)  that  a  real  solution  6^  will  be  modified  to  become 
6I-i(o7/2)(5Q/^6l)■^  and  hence  the  real  pole  for  which  dn/36i<0  is  moved  to 
the  upper  half  plane,  while  that  for  which  30/56,  >0  is  moved  to  the  lower 
half  plane.  Given  that  the  residue  at  such  a  pole  is  proportional  to  (30/36i)'\ 


^One  positive  and  one  negative.  These  solutions  will  have  the  form  ±6, 
for  an  orthotropic  system. 


1349 


it  follows  that  the  sign  of  the  residue  which  arises  from  the  contour  integral 
is  determined  by  the  integration  path  selected,  and  hence  by  the  sign  of 
These  considerations  lead  to  the  result 


~Tr~l  fi|3fi/3«,|(l+/)/) 


(12) 


where  e,(£2,M)  is  the  appropriate  solution  to  equation  (11).  The  evaluation  of 
the  integral  over  €3  is  discussed  in  the  following  section. 

3.3  Integration  over  €3 

Since  the  present  concern  is  with  the  response  of  the  system  at  some  distance 
from  the  excitation  point,  the  integral  over  e,  which  appears  in  equation  (12) 
can  be  evaluated  to  an  acceptable  degree  of  accuracy  by  using  the  method  of 
steepest  descent  [7],  With  this  approach  it  is  first  necessary  to  identify  the 
value  of  €2  for  which  the  exponent  -i(eirti+e2«2)  is  stationary.  The  condition 
for  this  is 

{3ejde^n^+n^=0.  (13) 


Now  Gi  and  satisfy  the  dispersion  relation,  equation  (11),  and  thus  equation 
(13)  can  be  re-expressed  in  the  form 

(aQ/a62)«r(5^2/a€,)«2=o,  (i4) 


where  it  has  been  noted  from  equation  (11)  that,  for  fixed  co,  3ei/3€2=- 
(5Q/3e2)/(9fi/36i),  In  the  absence  of  damping  the  wave  group  velocity  lies  in 
the  direction  (SQ/Sei  and  in  this  case  it  follows  from  equation  (14) 

that  the  group  velocity  associated  with  the  required  value  of  €2  is  along  («i  ru). 
For  light  damping  this  result  will  be  substantially  unaltered,  although  damping 
will  have  an  important  effect  on  the  value  of  the  exponent  -\{€^n^-\-e2iv^  at  the 
stationary  point.  This  effect  can  be  investigated  by  noting  initially  that 

d{e^n^+&^n^)IBr}~{deJbri  B&Jbr]).{n^  n^.  (15) 

Now  it  follows  from  equation  (11)  that  for  light  damping  {ri<l) 

(dQ/de,  dQlde,).{de,ldr]  de^/dv)  =  -io)l2,  (16) 


and  hence  equations  (14)-(16)  can  be  combined  to  yield  the  following  result 
at  the  stationary  point 

d{e^n^+e^n2)/dr)  =  -io)n/2c^.  (17) 


1350 


Here  Cg=^{{d£l!dex?+{d^lbe^'^]  is  the  resultant  group  velocity  and 
n=V[ni+ni-'\  is  the  radial  distance  (in  units)  from  the  excitation  point  to  the 
unit  under  consideration.  It  follows  that  in  the  immediate  vicinity  of  the 
stationary  point  the  exponent  can  be  expanded  in  the  form 


(18) 

where  the  subscript  0  indicates  that  the  term  is  to  be  evaluated  at  the 
stationary  point  under  the  condition  7]=0;  for  ease  of  notation,  this  subscript 
is  omitted  in  the  following  analysis.  The  method  of  steepest  descent  proceeds 
by  substituting  equation  (18)  into  equation  (12)  and  assuming  that:  (i)  the  main 
contribution  to  the  integral  arises  from  values  of  62  in  the  immediate  vicinity 
of  the  stationary  point;  (ii)  the  integrand  is  effectively  constant  in  this  vicinity, 
other  than  through  variation  of  the  term  e2-(e2)o  which  appears  in  equation 
(18);  (iii)  under  conditions  (i)  and  (ii)  the  integration  range  can  be  extended 
to  an  infinite  path  without  significantly  altering  the  result.  The  method  then 
yields  [7] 

w{n,x)  =  ~if  *F7o[20V|aQ/a£j/2xp(A:)p(A:o)|«i(aV5e2)|  . 


(19) 

where/ is  the  normalized  complex  wave  amplitude  which  appears  in  equation 
(9),  and  all  terms  are  to  be  evaluated  at  the  stationary  point. 

The  stationary  point  associated  with  equation  (19)  is  that  point  for 
which  the  group  velocity  is  in  the  {n^  rQ  direction.  Geometrically,  this  is  the 
point  at  which  the  normal  to  the  curve  a)=Q(ei,  £2)  itt  the  plane  lies  in  the 
{n^  Ho)  direction.  Three  such  curves  are  shown  schematically  in  Figure  4, 
together  with  a  specified  (n,  722)  direction.  For  the  frequencies  and  coj  the 
situation  is  straight  forward,  in  the  sense  that  a  unique  stationary  point  exists 
for  any  {n^  n^)  direction.  For  the  frequency  0J2  the  situation  is  more  complex, 
since:  (i)  two  stationary  points  occur  for  the  (n^  rQ  direction  shown,  and  (ii) 
no  stationary  point  exists  if  the  {n^  direction  lies  beyond  the  heading  B 
shown  in  the  figure  (the  dashed  arrow  represents  the  normal  with  maximum 
inclination  to  the  axis).  In  case  (i)  equation  (19)  should  be  summed  over 
the  two  stationary  points,  while  in  case  (ii)  the  method  of  steepest  descent 
predicts  that  w{n,x)  will  be  approximately  zero,  leading  to  a  region  of  very 
low  vibrational  response.  If  the  direction  («i  coincides  with  the  dashed 
arrow,  then  equation  (19)  breaks  down,  since  it  can  be  shown  that 
at  this  point.  The  heading  indicated  by  the  dashed  arrow  represents  a  caustic 
[7],  and  the  theory  given  in  the  present  section  must  be  modified  for  headings 


1351 


(Wi  Wj)  which  are  in  the  immediate  vicinity  of  the  caustic  -  full  details  of  the 
appropriate  modifications  are  given  in  reference  [4].  An  example  which 
illustrates  the  application  of  equation  (19)  is  given  in  section  4. 

3.  RESPONSE  TO  AN  IMPULSIVE  POINT  LOAD 

If  the  system  is  subjected  to  an  impulsive  (i.e.  a  delta  function  applied  at 
r=0),  rather  than  harmonic,  point  load,  then  equation  (10)  becomes  [8] 


w{n,x,t)={N^NJ2Tp)  f  [  g* 

II  (20) 


where  co  =  ^](€i,€2)-  The  method  of  stationary  phase  can  be  applied  to  this 
expression  to  yield  [8] 

w{n,x,t)-{\l2yr)[p{x)p{x)\J\r‘^^^^ 

exp(-/Gjtti  -k^n^nQt+ib) } , 


(21) 

where  all  terms  are  evaluated  at  the  stationary  point,  and  J  and  5  are  defined 
as 


/=(a"n/a6?)(a"Q/ae^-(3"0/ae,a6,)",  S=(Tr/4)sgn(a%/fle?){l+sgn(/)}. 

(22,23) 

In  this  case  the  stationary  point  is  given  by  the  solution  to  the  equations 

=(afi  =(30  /3e,>.  (24,25) 

In  practice  equations  (24)  and  (25)  may  yield  multiple  solutions  (stationary 
points),  in  which  case  equation  (21)  should  be  summed  over  all  such  points. 
Furthermore,  stationary  points  having  7=0  indicate  the  occurrence  of  a 
caustic,  and  equation  (21)  must  be  modified  in  the  immediate  vicinity  of  such 
points  as  detailed  in  reference  [8].  An  example  of  the  application  of  equation 
(21)  is  given  in  the  following  section. 


1352 


4.  EXAMPLE  APPLICATION 


4. 1  The  System  Considered 

The  foregoing  analysis  is  applied  in  this  section  to  a  two-dimensional  periodic 
structure  which  consists  of  a  rectangular  grid  of  lumped  masses  m  which  are 
coupled  through  horizontal  and  vertical  shear  springs  of  stiffness  and  h 
respectively.  Each  mass  has  a  single  degree  of  freedom  consisting  of  the  out- 
of-plane  displacement  w,  and  a  linear  spring  of  stiffness  k  is  attached  between 
each  mass  and  a  fixed  base.  It  is  readily  shown  that  the  system  has  a  single 
phase  constant  surface  of  the  form 

Q  -(ep€,)=iLii(l-cos€i)+/x2(l-cos€2)+a;^, 


where  ix^^lkjm,  and  o)^-==klm.  The  function  U  can  be  used  in 

conjunction  with  the  analysis  of  the  previous  sections  to  yield  the  response  of 
the  system  to  harmonic  and  impulsive  point  loading;  in  this  regard  it  can  be 
noted  that  for  the  present  case  p{x)=m,  V—l,  and /(:»:)  =  1. 

4.2  Response  to  Harmonic  Loading 

The  surface  is  shown  as  a  contour  plot  in  Figure  5  for  the  case 

m=1.0,  oj„==0,  Ati  =  1.0,  Results  for  the  forced  harmonic  response 

of  this  system  at  the  two  frequencies  w  =  1.003  and  cx>  =  1.181  are  shown  in 
Figures  6  and  7.  In  each  case  the  response  of  a  40  x40  array  of  point  masses 
is  shown;  a  unit  harmonic  point  load  is  applied  to  mass  (21,21)  and  the  loss 
factor  is  taken  to  be  =0.05.  Two  sets  of  contours  are  shown  in  each  Figure: 
the  smooth  contours  have  been  calculated  by  using  equation  (19)  while  the 
more  irregular  contours  have  been  obtained  by  a  direct  solution  of  the 
equations  of  motion  of  the  finite  1600  degree-of-freedom  system.  By 
considering  the  results  shown  in  Figure  6,  it  can  be  concluded  that:  (i)  for  the 
present  level  of  damping  the  finite  system  effectively  behaves  like  an  infinite 
system,  and  (ii)  the  analytical  result  yielded  by  equation  (19)  provides  a  very 
good  quantitative  estimate  of  the  far  field  response.  It  can  be  noted  from 
Figure  5  that  no  caustic  occurs  for  a)  =  1.003,  in  the  sense  that  equation  (14) 
yields  only  one  stationary  point  which  contributes  to  equation  (19).  In 
contrast,  a  caustic  does  occur  for  the  case  a)  =  1.181,  and  this  leads  to  the  very 
irregular  spatial  distribution  of  response  shown  in  Figure  7.  Two  stationary 
points  contribute  to  equation  (19),  and  constructive  and  destructive 
interference  between  these  contributions  is  responsible  for  the  rapid 
fluctuations  in  the  response  amplitude.  It  is  clear  that  the  response  exhibits 
a  "dead  zone"  for  points  which  lie  beyond  the  caustic  heading  (in  this  case 
30.25°  to  the  «i-axis),  as  predicted  by  the  analysis  presented  in  section  2. 

4.3  Response  to  Impulsive  Loading 

The  impulse  response  of  a  system  having  m=1.0,  /xi  =  1.0,  ^2=0. 51,  and 


1353 


co„'“0.25  has  been  computed.  The  impulse  was  taken  to  act  at  the  location 
/Zi=«2=0  and  the  time  history  of  the  motion  of  each  mass  in  the  region  - 
iO<(72i,n2)  ^  10  was  found  by  using  equations  (21)-(23).  For  each  mass  the 
maximum  response  \w\  was  recorded,  and  the  results  obtained  are  shown  as 
a  contour  plot  in  Figure  8.  In  accordance  with  Fourier’s  Theorem,  the 
impulse  response  of  the  system  contains  contributions  from  all  frequencies, 
and  therefore  the  spatial  distribution  of  |  w|  can  be  expected  to  lie  somewhere 
between  the  two  extreme  forms  of  harmonic  response  exhibited  in  Figures  6 
and  7.  This  is  in  fact  the  case,  and  the  response  shown  in  Figure  8  retains  a 
distinctive  spatial  pattern.  As  discussed  in  reference  [8],  the  results  shown  in 
Figure  8  are  in  good  agreement  with  direct  simulation  of  the  impulse  response 
of  the  system. 


5.  CONCLUSIONS 

This  paper  has  considered  the  response  of  a  two-dimensional  periodic  structure 
to  both  harmonic  and  impulsive  point  loading.  With  regard  to  harmonic 
loading,  it  has  been  shown  that  the  spatial  pattern  of  the  response  is  strongly 
dependent  on  the  occurrence  of  a  caustic:  if  no  caustic  occurs  then  the 
response  has  a  fairly  smooth  spatial  distribution,  whereas  the  presence  of  a 
caustic  leads  to  an  irregular  spatial  distribution  and  a  "dead  zone"  of  very  low 
response.  This  type  of  feature  is  also  exhibited,  although  to  a  lesser  degree, 
in  the  spatial  distribution  of  the  response  to  an  impulsive  point  load.  This 
behaviour  could  possibly  be  exploited  to  reduce  vibration  transmission  along 
a  specified  path,  although  the  practicality  of  this  approach  for  a  complex 
system  has  yet  to  be  investigated.  The  present  analytical  approach  can  be 
applied  to  all  types  of  two-dimensional  periodic  structure  -  the  information 
required  consists  of  the  phase  constant  surface(s)  0(61,62)  and  the  associated 
wave  form(s)  f{x),  both  of  which  are  yielded  by  standard  techniques  for  the 
analysis  of  free  wave  motion  in  periodic  structures  [1,2]. 

REFERENCES 

1.  S.S.  MESTER  and  H.  BENAROYA  1995  Shock  and  Vibration  2,  69- 
95.  Periodic  and  near-periodic  structures. 

2.  D.J.  MEAD  1996  Journal  of  Sound  and  Vibration  190,  495-524. 
Wave  propagation  in  continuous  periodic  structures:  research 
contributions  from  Southampton  1964-1995. 

3.  L.  MEIROVITCH  1986  Elements  of  Vibration  Analysis,  Second 
Edition.  New  York:  McGraw-Hill  Book  Company. 

4.  R.S.  LANGLEY  1996  Journal  of  Sound  and  Vibration  (to  appear). 
The  response  of  two-dimensional  periodic  structures  to  point  harmonic 
forcing. 


1354 


1 

■ 

■ 

■ 

1 

■ 

■ 

■ 

0 

m 

■ 

1 

■ 

Xj.  ■j' 

Oi 

< - 

^1 

1  ^ 

Figure  I ,  Schematic  of  a  two-dimensional  periodic  structure.  The  arrow  indicates'  the  reference  unit 
(with  n=0)  while  the  circle  represents  a  general  point  (re,x).  The  structure  may  have  a  third  spatial 
coordinate  X3,  which  for  convenience  is  not  shown  in  the  figure.  The  point  load  considered  in  section 
3  is  applied  at  the  location  of  the  arrow. 


Figure  2.  Phase  constant  surfaces  for  a  plate  which  rests  on  a  square  grillage  of  simple  supports. 
Q  is  a  non-dimensional  frequency  which  is  def  as  Q=a)LV(m/D),  where  m  and  D  are  respectively 
the  mass  per  unit  area  and  the  flexural  rigidii^  the  plate,  and  L  is  the  support  spacing. 


1355 


Figure  5.  Contour  plot  of  the  phase  constant  surface  for  the  case  ^=0.57.  The  contours 

are  separated  by  an  increment  Aco =0. 1477.  The  two  contours  considered  in  section  4.2  are  indicated 
as  follows:  (a)  a)  =  1.033;  (b)  a)  =  1.181. 


Figure  6.  Response  |H'(n,x)|2  of  the  40x4v  .uass/spring  system  to  a  unit  harmonic  force  of 
frequency  w- 1.033  applied  at  the  location  i=y=21.  The  contours  correspond  to  the  response  levels 
lK«,x)!'=0.01,  0.02,  andO.05. 


1357 


STICK-SLIP  MOTION  OF  AN  ELASTIC  SLIDER  SYSTEM  ON 
A  VIBRATING  DISC 


HOuyang  J  E  Mottershead  M  P  CartmeU '  MIFiiswell^ 
Department  of  Mechanical  Engineering,  University  of  Liverpool 
^  Department  of  Mechanical  Engineering,  University  of  Edinburgh 
^  Department  of  Mechanical  Engineering,  University  of  Wales  Swansea 

ABSTRACT 


The  in-plane  vibration  of  a  slider-mass  which  is  driven  around  the  surface  of  a 
flexible  disc,  and  the  transverse  vibration  of  the  disc,  are  investigated.  The  disc 
is  taken  to  be  an  elastic  annular  plate  and  the  slider  has  flexibility  in  the 
circumferential  (in-plane)  and  transverse  directions.  The  static  fiiction 
coefi&cient  is  assumed  to  be  higher  than  the  kinetic  friction.  As  a  result  of  the 
fiiction  force  acting  between  the  disc  and  the  slider  system,  the  slider  will 
oscillate  in  the  stick-slip  mode  in  the  plane  of  the  disc.  The  transverse  vibration 
induced  by  the  slider  will  change  the  normal  force  of  the  slider  system  acting 
on  the  disc,  which  in  turn  will  change  the  in-plane  oscillation  of  the  slider.  For 
different  values  of  system  parameters,  the  coupled  in-plane  oscillation  of  the 
slider  and  transverse  vibration  of  the  disc  will  exhibit  quasi-periodic  as  well  as 
chaotic  behaviour.  Rich  patterns  of  chaotic  vibration  of  the  slider  system  are 
presented  in  graphs  to  illustrate  the  special  behaviour  of  this  non-smooth 
nonlinear  dynamical  system  The  motivation  of  this  work  is  to  analyse  and 
understand  the  instability  and/or  squeal  of  physical  systems  such  as  car  brake 
discs  where  there  are  vibrations  induced  by  non-smooth  dry-fiiction  forces. 


NOMENCLATURE 


a ,  b  mner  and  outer  radii  of  the  annular  disc 
c  damping  coefficient  of  the  slider  in  in-plane  direction 
h  thickness  of  the  disc 


i  =  V=T 


1359 


k ,  k  transverse  and  in-plane  stiffiiess  of  the  slider  system 


m 

r 


t 


^stiek 


mass  of  the  slider 

radial  co-ordinate  in  cylindrical  co-ordinate  system 
radial  position  of  the  slider 

modal  co-ordinate  for  k  nodal  circles  and  /  nodal  diameters 
time 

the  time  of  the  onset  of  sticking 
u ,  Mp  transverse  displacement  of  the  slider  mass  and  its  initial  value 
w ,  Wq  transverse  displacement  of  the  disc  and  its  initial  value 
flexural  rigidity  of  the  disc 
Kelvin-type  damping  coefficient 
Young’s  modulus 

initial  normal  load  on  the  disc  jfrom  the  slider  system 
total  normal  force  on  the  disc  from  the  slider  system 
combination  of  Bessel  functions  representing  mode  shape  in  radial 
direction 

circumferential  co-ordinate  of  cylindrical  co-ordinate  system 

kinetic  and  static  dry  fiction  coefficient  between  the  shder  and  the 


D 

D' 

E 

N 

P 

R. 


0 


disc 


V 

P 

(P 

^  stick 
¥ 
¥ki 


CO. 


Poisson’s  ratio 

damping  ratio  of  the  disc 

specific  density  of  the  disc 

absolute  circumferential  position  of  the  slider 

absolute  circumferential  position  of  the  slider  when  it  sticks  to  the  disc 
circumferential  position  of  the  slider  relative  to  the  drive  point 
mode  function  for  the  transverse  vibration  of  the  disc  corresponding  to 

^kl 

natural  (circular)  frequency  correq)onding  to  <5^^ 


1360 


Q  constant  rotating  speed  of  the  drive  point  around  the  disc  in  radians  per 
second 


INTRODUCTION 


There  exists  a  whole  class  of  mechanical  systems  which  involve  discs  rotating 
relative  to  stationery  parts,  such  as  car  brake  discs,  clutches,  saws,  computer 
discs  and  so  on.  In  these  systems,  dry-friction  induced  vibration  plays  a  crucial 
role  in  system  performance.  If  the  vibration  becomes  excessive,  the  system 
might  fail,  or  cease  to  perform  properly,  or  make  offensive  noises.  In  this 
paper,  we  investigate  the  vibration  of  an  m-plane  slider  system,  with  a 
transverse  mass- spring- damper,  attached  through  an  in-plane  spring  to  a  drive 
point  which  rotates  at  constant  speed  around  an  elastic  disc,  and  the  vibrations 
of  the  disc.  Dry  friction  acts  between  the  sHder  system  and  the  disc. 

Dry-friction  induced  vibration  has  been  studied  extensively  [1-4].  For  car  brake 
vibration  and  squeal,  see  the  review  papers  [5,6].  The  stick-slip  phenomenon  of 
dry-friction  induced  vibration  is  studied  in  the  context  of  chaotic  vibration  [7- 
10].  Popp  and  Stelter  [7]  studied  such  motion  of  one  and  two  degrees  of 
freedom  system  and  foimd  chaos  and  bifiircation.  They  also  conducted 
experiments  on  a  beam  and  a  circular  plate  (infinite  number  of  degrees  of 
freedom).  These  theoretical  works  are  about  systems  of  less  than  three  degrees 
of  freedom,  and  the  carrier  which  activates  the  friction  is  assumed  to  be  rigid. 
In  this  paper,  we  consider  an  elastic  disc  so  that  the  transverse  vibrations  of  the 
disc  are  important.  As  a  result  of  including  the  transverse  vibrations  of  the  disc, 
rich  patterns  of  chaos,  which  have  not  been  reported  previously  are  found.  If 
there  is  only  shding  present  at  constant  speed,  the  problem  is  reduced  to  a 
linear  parametric  analysis  which  was  carried  out  for  a  pin-on- disc  system  in 
[11]  and  for  a  pad-on-disc  system  in  [12,13]. 

m-PLANE  OSCILLATION  OF  THE  SLIDER  SYSTEM 


As  the  drive  point,  which  is  connected  to  the  shder-mass  through  an  in-plane, 
elastic  spring,  is  rotated  at  constant  angular  speed  around  the  disc,  the  driven 
slider  will  undergo  stick-slip  oscillations.  The  whole  system  of  the  shder  and 
the  disc  is  shown  in  Figure  1. 

The  equation  of  the  in-plane  motion  of  the  slider  system  relative  to  the  rotating 
drive  point,  in  the  sliding  phase,  is. 


1361 


(1) 


while  in  sticking,  the  equation  of  the  motion  becomes. 

The  relationship  between  the  relative  motion  of  the  slider  system  to  the  drive 
point  and  its  absolute  motion  (relative  to  the  stationary  disc)  is 

(p  =  Qt  +  y/,  (3) 

We  consider  the  foEowing  initial  conditions  which  are  intended  to  simulate 
what  happens  in  a  disc  brake.  The  slider  system  is  at  rest  and  there  is  no  normal 
loading  on  the  disc  j&om  the  slider.  Then  a  constant  normal  load  is  applied 
which  causes  transverse  vibrations  in  the  disc.  At  the  same  time,  the  drive  point 
is  given  a  constant  angular  velocity.  Other  initial  conditions  are  possible,  so 
that  there  is  no  loss  of  generality. 

First,  sliding  from  the  initial  sticking  phase  occurs  when, 

The  slider  will  stick  to  the  disc  agaiu  when, 

ju^P  (during  sliding),  (5) 

or  it  will  begin  to  sHde  again  iJ^ 

ij/  =  ,  \k^rQtp-\  >  (dxiring  sticking).  (6) 

Consequently,  the  slider  system  will  stick  and  slide  consecutively  on  the  disc 
surface. 


TRANSVERSE  VIBRATION  OF  THE  ANNULAR  DISC 

The  equation  of  motion  of  the  disc  under  the  slider  system  is, 

ph^  +  D'V^'w  +  DW  =  --5(r-r„)5i$-<p)P.  (7) 

a/"  r 

The  total  force  P  is  the  summation  of  initial  normal  load  N  and  the  resultant 
of  the  transverse  motion  w  of  the  slider.  Its  expression  is, 

P  -  N +  mu  +  cu-h kiu-u^) .  (8) 

Since  it  is  assumed  that  the  slider  system  is  always  in  perfect  contact  with  the 
disc,  then, 


1362 


(9) 


u{t)  =  w{r„(p{t)A- 

Substitution  of  ecpiations  (8)  and  (9)  into  (7)  leads  to. 


ph 


^  —  +  /:)VV  =  --5(r  -  r,  )6(^  -  (p)[N  + 


dt 


dt 


^..dw  d^w  d^w  .dw  dw 


(10) 


k(w-wj\. 

Note  that  equation  (10)  is  valid  whether  the  slider  system  is  sticking  or  sliding. 
When  the  slider  sticks  to  the  disc,  equation  (10)  reduces  to. 


+  D-V‘  ^ +£)W  =  --5(r  - )5(5 -<p)x 

St  St  r  (11) 

d'^W  ^  M 

[A^+m— +  c— +  A:(w-w„)]. 


COUPLED  VIBRATIONS  OF  THE  SLIDER  AND  THE  DISC 


Assume  that  the  transverse  motion  of  the  disc  can  be  represented  by, 

M>{r,e,t)=ttii/,{r,0)q,it),  (12) 

Jt=0  /=-<io 


and, 

where  {r)  is  a  combination  of  Bessel  functions  satisfying  the  boundary 
conditions  in  radial  direction  at  the  inner  radius  and  outer  radius  of  the  disc. 
The  modal  functions  satisfy  the  ortho-normality  conditions  ofj 


(14) 


1363 


Equations  (10)  and  (11)  can  be  simplified  by  being  written  in  terms  of  the 
modal  co-ordinates  from  equation  (12). 

During  sticking,  the  motion  of  the  whole  system  of  the  slider  and  the  disc  can 
be  represented  by, 

N 

+  2^0) - 
— Z^„('-o)^«('-o)exp[i(5-0(»]x  (15) 

ph.U  ''=0  s=-«o 

{mq„+cq^+k{q^-q„{^)\). 


The  sticking  phase  can  be  maintained  i^ 


rA\¥\</^XJ^+- 


^Z  Z  ^«(?')exp(i/?»)x 


{mq„+cq„+  k[q„  -  (0)] }  ] . 

While  in  sliding,  the  motion  of  the  whole  system  can  be  represented  by. 


%+2^a>^,qu+a>lq„  = 


^^«('i)exp(-i/<»)- 


-rrri  Z^„('-o)^«(n,)exp[i(5-0«!’]x 

pnO  r=<i  5^=-oo 

H?,.  +i2sw„  +isw„)  + 

k{q„-qS'^)]}, 


r„{m{j/  +  kw)  =  -/i,Sign((Z))[A''  +- 


^Z  Z  KMx 


exp(LS(Z>)M?„  +i2s^„  +(ls^-5>^)9„]  + 

c{q„  +>s(?g„) +*[!?„  -?„(o)]}]. 

The  sliding  phase  can  be  maintained  if, 


1364 


\K^M< mXN +-J^t  t  K{r,)es.p{is(p)x 
■yjphb  ^=0^^ 

{iTiq„+i2sj>q„+(is^-s^j)^)qJ+  (20) 

c(?„  +isj^J  +  k[q„  -9„(0)]}], 

when, 

\j/  =  ~0  or  ^  =  0.  (21) 

COMPUTING  PROCESS 


As  the  shder  system  sticks  and  sHdes  consecutively,  the  governing  equations  of 
the  coupled  motions  of  the  whole  system  switch  repeatedly  from  equations 
(15),  (16)  and  (17)  to  equation  (18),  (19)  and  (20).  TTie  system  is  not  smooth. 
Since  the  condition  which  controls  the  phases  of  the  slider  system  itself 
depends  on  the  motions,  it  is  also  a  nonlinear  system,  whether  is  a  constant 
or  a  function  of  relative  speed  ^ .  In  order  to  get  modal  co-ordinates,  we  have 
to  truncate  the  mfinite  series  in  equation  (12)  to  jfinite  terms.  Then  numerical 
integration  is  used  to  solve  equations  (15),  (18)  and  (19).  Here  a  fourth  order 
Rimge-Kutta  method  is  used  for  second  order  simultaneous  ordinary 
differential  equations. 

Since  equation  (18)  has  time-dependent  coefficients,  time  step  length  has  to  be 
very  small.  Constant  time  step  lengths  are  chosen  when  the  m-plane  slider 
motion  is  well  within  the  sticking  phase  or  the  shding  phase  in  the  numerical 
integration.  As  it  is  imperative  that  the  time  step  should  be  chosen  such  that  at 
the  end  of  some  time  intervals  the  shder  happens  to  be  on  the  sticking- shding 
interfaces,  we  use  a  prediction  criterion  to  choose  next  time  step  length  when 
approaching  these  interfaces.  Therefore,  at  the  sticking-shding  interfaces,  the 
time  step  length  is  variable  (actuahy  smaher  than  it  is  while  weh  within  sticking 
or  shding).  Nevertheless,  tbe  interfaces  equations  (  equation  (17)  or  equations 
(20-21) )  are  only  approximately  satisfied  [10]. 

When  transverse  motion  of  the  shder  system  becomes  so  violent  that  the  total 
normal  force  P  in  it  becomes  negative  or  becomes  several  times  larger  than 
the  initial  normal  load  7/ ,  we  describe  the  system  as  being  unstable.  Then  the 
motion  begins  to  diverge.  But  this  instabihty  should  be  distinguished  from  a 
chaotic  motion  which  is  bounded  but  never  converge  to  a  point. 


1365 


NUMERICAL  EXAMPLES 


The  following  data  are  used  in  the  computation  of  numerical  examples: 
a  =  0.065m,  b  =  012m,  OTm, /z  =  0.001m;  =  120GPa,  v=  0.35, 

Z)*=  0.00004;  yW,  =0.4,  //^=0.24,  A:  =  lOOON/m,  =  lOON/m, 

m  =  0.1kg,  p  =  7000kg / m^  The  disc  is  clamped  at  inner  radius  and  free  at 

outer  radius.  Note  that  in  these  numerical  examples,  the  disc  thickness  is 
dehherately  taken  to  be  very  small  in  order  to  reduce  the  amount  of  computing 
work.  However,  this  will  not  affect  the  qualitative  features  of  the  results  or 
conclusion  drawn  from  the  results  thus  obtained.  The  first  five  natural 
(circular)  frequencies  are  451.29,  462.73,  426.73,  508.23,  508.23.  We  will 
concentrate  on  the  vibration  solutions  at  different  levels  of  initial  normal  load. 
But  occasionally  solutions  at  different  rotating  speed  or  different  damping 
ratios  are  investigated.  Unless  specified  expressly,  the  Poincare  sections  are  for 
the  in-plane  vibration  of  the  slider  system 

First  of  all,  we  study  the  effect  of  the  normal  loadA^.  Take  f2=l0  and 
^ —  When  N  is  very  small^  the  Poincare  section  is  a  perfect  ellipse 

which  indicate  the  in-plane  vibration  of  the  slider  system  is  quasi-periodic,  as 
the  transverse  vibration  of  the  disc  is  too  small  to  affect  total  normal  force  P . 
A  typical  plot  of  such  motion  is  shown  in  Figure  2  for  A^=0.5kPa.  As  N 
increases,  the  sticking  period  gets  longer,  the  bottom  part  of  the  elhpse  evolves 
into  a  straight  line,  indicating  phase  points  within  the  sticking  phase.  One  of 
such  plots  is  given  in  Figure  3  for  N  =3kPa.  A  further  increase  of  N  not  only 
lengthens  the  straight  line  part  of  the  Poincare  section,  but  also  creates  an 
increasingly  ragged  outline  in  the  arch  part  of  the  plot.  The  curve  is  no  longer 
smooth  and  it  seems  that  the  in-plane  motion  begins  to  enter  a  chaotic  state 
from  the  quasi-periodic  state.  Figure  4  presents  the  Poincare  section  plot  for 
N  =7.5kPa.  There  is  a  transition  period  from  quasi-periodic  motion  to  chaotic 
motion,  extending  from  N  =6kPa  up  to  N  =9kPa.  Chaos  becomes  detectable 
at  iV=10kPa,  whose  Poincare  section  is  shown  in  Figure  5.  Then  chaotic 
vibration  follows.  When  N  ~15kPa,  the  arch  part  of  the  Poincare  section 
becomes  so  fuzzy  and  thick  that  it  should  no  longer  be  considered  as  a  curve, 
but  rather  a  narrow  (fractal)  area.  A  hlow-up’  view  of  the  arch  part  reveals 
that  phase  points  are  distributed  across  the  arch.  Both  plots  are  ^own  in 
Figure  6.  Between  iV'=17.ikPa  and  18.325kPa,  the  vibration  of  the  slider 
enters  a  new  stage,  with  Poincare  sections  looking  like  star  clusters  as 
illustrated  in  Figures  7  and  8.  This  kind  of  motions  are  rather  extraordinary  and 


1366 


have  not  been  reported  in  other  works  on  stick-shp  motions  with  a  rigid 
carrier.  Afterwards,  the  ‘arch-door’  hke  Poincare  sections  come  back  (see 
Figure  9).  The  difference  from  previous  Poincare  sections  of  lower  N  is  that 
the  new  Poincare  sections  look  like  overlapping  of  earlier  Poincare  sections, 
which  indicates  a  clear  layered  structure,  as  diown  in  Figure  10  and  more 
obviously  in  the  left  hand  side  of  Figure  11.  At  this  stage,  the  vibration  is  very 
chaotic.  To  give  the  reader  a  better  picture,  the  Poincare  section  of  a  fixed 
point  on  the  disc  at  ( =  0.1m  and  =  0 ),  is  also  shown  in  the  right  hand  side 
of  Figure  11.  The  Poincare  sections  of  the  slider-mass  and  a  point  on  the  disc 
are  also  given  in  Figures  12-15.  In  Figure  12  for  A^=30.5kPa,  the  vibration 
goes  unstable.  Here  again,  the  Poincare  sections  have  not  been  reported 
elsewhere. 

If  disc  damping  is  increased,  vibration  will  become  more  regular,  as  shown  in 
Figure  13.  Comparing  Figures  11  with  13,  we  see  that  increase  of  disc  damping 
makes  the  vibration  more  concentrated  though  not  always  smaller.  Unstable 
vibration  can  be  stabilised  with  more  disc  damping,  as  seen  from  Figure  14. 

If  there  is  no  damping  at  aU,  the  resulting  vibration  due  to  dry  fiiction  will  be 
unstable,  even  at  very  small  normal  load  N .  In  Figure  15,  the  motion  of  the 
slider  tends  to  run  away  in  the  tangential  direction  from  the  normal  ellipse 
attractor,  while  the  motion  of  the  disc  goes  unbounded. 

Increasing  the  speed  of  the  drive  point  seems  to  make  vibration  more  chaotic 
and  more  unstable,  as  shown  in  Figures  16-18.  At  this  stage,  however,  we  are 
unable  to  make  a  definite  conclusion  on  rotating  speed  as  there  might  be 
intervals  of  regular  motions  and  intervals  of  chaotic  motion  for  .  More 
numerical  examples  must  be  computed  to  draw  a  positive  conclusion  on  this 
parameter. 

The  correlation  dimension  is  not  a  good  measure  of  the  vibration  for  the 
current  problem  because  its  values  fluctuate  in  some  numerical  examples.  This 
failure  was  perhaps  first  discovered  in  [7].  The  reason  can  be  either  that  the 
system  is  non-smooth,  or  that  the  system  has  multiple  degrees  of  freedom,  or 
both.  Therefore,  the  correlation  dimension  or  any  other  fractal  dimensions  is 
not  presented  in  this  paper. 


CONCLUSIONS 


In  this  paper,  we  studied  the  in-plane  stick-slip  vibration  of  a  slider  system  with 
a  transverse  mass-spring-damper  driven  around  an  elastic  disc  through  a  spring 
from  a  constant  speed  drive  point,  and  transverse  vibrations  of  the  disc.  The 


1367 


whole  system  had  been  reduced  to  six  degrees  of  freedom  after  simplification. 
From  numerical  examples  computed  so  far,  we  can  conclude  that: 

1.  Both  vibrations  are  very  complex  as  this  is  a  multi-degree  of  freedom,  non¬ 
smooth  system  Rich  patterns  of  chaotic  vibration  are  found.  Some  have  not 
been  reported  elsewhere. 

2.  For  the  normal  pressure  parameter,  smaller  values  allow  quasi-peiiodic 
solutions.  Greater  pressures  result  in  chaotic  motions.  At  certain  large 
pressures,  the  vibrations  become  unstable. 

3.  Disc  dartping  makes  vibration  more  concentrated  to  smaller  areas  and  when 
sufficiently  large  it  can  stabilise  otherwise  unstable  vibration. 

4.  An  increase  in  the  rotating  speed  can  make  the  vibration  more  chaotic  or 
more  unstable. 

5.  Correlation  dimension  is  not  a  good  measure  of  the  vibration  of  this  multi¬ 
degree  of  freedom,  non-smooth  dynamical  system 

6.  Much  more  investigation  needs  to  be  earned  out  in  understanding  and 
characterising  the  vibration  of  multi- degree  of  freedom,  non- smooth  dynamical 
systems. 


ACKNOWLEDGEMENT 

This  research  is  supported  by  the  Engineering  and  Physical  Sciences  Research 
Council  (grant  niunber  J35177)  and  BBA  Friction  Ltd. 

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10.  Wiercigroch,  M.,  A  note  on  the  switch  function  for  the  stick-slip 
phenomenon.  J.SoundVib.,  1994, 175,  700-4 

11.  Chan,  S.N.,  Mottershead,  J.E.  and  Cartmell,  M.P.,  Parametric  resonances 
at  subcritical  speeds  in  discs  with  rotating  frictional  loads.  Proc.  Instn. 
Mech.  Engrs,  1994,  208,  417-25 

12.  Mottershead,  J.E.,  Ouyang,  R,  Cartmell,  M.P.  and  Friswell,  M.I., 
Parametric  Resonances  in  an  annular  disc,  with  a  rotating  system  of 
distributed  mass  and  elasticity;  and  the  effects  of  friction  and  dan:q)ing. 
Proc.  Royal  Soc.  Lond.  A.,  1997,  453,  1-19 

13.  Ouyang,  H.,  Mottershead,  J.E.,  Friswell,  M.l.  and  Cartmell,  M.P.,  On  the 
prediction  of  squeal  in  automotive  brakes.  Proceedings  of  the  14th  Modal 
Analysis  Conference,  The  Society  of  Experimental  Mechanics,  Inc.,  1996, 
pp.  1009-16 


Figure  1.  Slider  system  and  disc  in  cylindrical  co-ordinate  system 


1369 


Figure  9.  iV=21kPa 


Figure  10.  iV=24kPa 


1374 


A  Finite  Element  Time  Domain 
Multi-Mode  Method  For  Large  Amplitude 
Free  Vibration  of  Composite  Plates 

Raymond  Y.  Y.  Lee,  Yucheng  Shi  and  Chuh  Mei 
Department  of  Aerospace  Engineering 
Old  Dominion  University,  Norfolk,  VA  23529-0247 


Abstract 

This  paper  presents  a  time-domain  modal  formulation  using  the  finite  element  method  for 
large-amplitude  firee  vibrations  of  generally  laminated  thin  composite  rectangular  plates.  Accurate 
fi'equency  ratios  for  fundamental  as  well  as  higher  modes  of  composite  plates  at  various  maximum 
deflections  can  be  determined.  The  selection  of  the  proper  initial  conditions  for  periodic  plate 
motions  is  presented.  Isotropic  beam  and  plate  can  be  treated  as  special  cases  of  the  composite 
plate.  Percentage  of  participation  from  each  linear  mode  to  the  total  plate  deflection  can  be 
obtained,  and  thus  an  accurate  frequency  ratio  using  a  minimum  number  of  linear  modes  can  be 
assured.  Another  advantage  of  the  present  finite  element  method  is  that  the  procedure  for  obtaining 
the  modal  equations  of  the  general  DujB5ng-type  is  simple  when  compared  with  the  classical 
continuum  Galerkin’s  approach.  Accurate  frequency  ratios  for  isotropic  beams  and  plates,  and 
composite  plates  at  various  amplitudes  are  presented. 


Introduction 

Large  amplitude  vibrations  of  beams  and  plates  have  interested  many 
investigators  [1]  ever  since  the  first  approximate  solutions  for  simply  supported 
beams  by  Woinowsky-Kiieger  [2]  and  for  rectangular  plates  by  Chu  and  Herrmann 
[3]  were  presented.  Singh  et  al.  [4]  gave  an  excellent  review  of  various  formulation 
and  assumptions  ,  including  the  finite  element  method  for  large  amplitude  firee 
vibration  of  beams.  Srirangaraja  [5]  recently  presented  two  alternative  solutions, 
based  on  the  method  of  multiple  scales  (MMS)  and  the  ultraspherical  polynomial 
approximation  (UP A)  method,  for  the  large  amplitude  firee  vibration  of  a  simply 
supported  beam.  The  fi'equency  ratios  for  the  fundamental  mode,  ca/C0L>  at  the  ratio 
of  maximum  beam  deflection  to  radius  of  gyration  of  5.0  (Wmax/r  =5.0)  are  3.3438 
and  3.0914,  using  the  MMS  and  the  UPA  method,  respectively.  Eleven  firequency 
ratios  including  nine  firom  reference  [4]  were  also  given  (see  Table  1  of  reference 
[5]).  It  is  rather  surprising  that  the  firequency  ratio  for  the  fundamental  mode  at 
Wn«x/r  =5.0  for  a  simply  supported  beam  varied  in  a  such  wide  range:  fi-om  the 
lowest  of  2.0310  to  the  highest  of  3.3438,  and  with  the  elliptic  function  solution  by 
Woinowsky-Kiieger  [2]  giving  2.3501.  Similar  wide  spread  exists  for  the  vibration 
of  plates.  Rao  et  al  [6]  presented  a  finite  element  method  for  the  large  amplitude 
firee  flexural  vibration  of  unstiffened  plates.  For  the  simply  supported  square  plate. 


1375 


frequency  ratios  from  six  different  approaches  were  reported  (see  Table  1  of 
reference  [6]).  The  frequency  ratio  at  Wmax/h  =1.0  varied  from  a  low  of  1.2967  to  a 
high  of  1.5314,  with  Chu  and  Herrmann’s  analytical  solution  [3]  at  1.4023. 

This  paper  presents  a  finite  element  time  domain  modal  formulation  for  the  large 
amplitude  free  vibration  of  composite  plates.  The  formulation  is  an  extension  from 
the  isotropic  plates  [7],  and  the  determination  of  initial  conditions  for  periodic 
motions  was  not  employed  in  reference  [7].  The  convergence  of  the  fundamental 
frequency  ratio  is  investigated  for  a  simply  supported  beam  and  a  simply  supported 
square  plate  with  a  varying  number  of  finite  elements  and  a  varying  number  of 
linear  modes.  Accurate  frequency  ratios  for  fundamental  and  higher  modes  at 
various  maximum  deflections,  and  percentages  of  participation  from  various  linear 
modes,  are  obtained  for  beams  and  composite  plates. 


Formulation 

Strain-Displacement  and  Constitutive  Relations 

The  von  Karman  strain-displacement  relations  are  applied.  The  strains  at  any 
point  z  through  the  thickness  are  the  sum  of  membrane  and  change  of  curvature 
strain  components: 


{e}  = 


-w.xx 

V,y 

>+< 

/  2 

>  +  z< 

-W,yy 

U,y+V,x^ 

W,yj 

=  {Em}  +  {sb}  +  z{K} 


(1) 


where  and  {eb}are  the  membrane  strain  components  due  to  in-plane 

displacements  u  and  v  and  the  transverse  deflection  w,  respectively.  The  stress 
resultants,  membrane  force  {N}  and  bending  moment  {M},  are  related  to  the  strain 
components  as  follows: 


■[A]  IB]' 
[B1  [Di; 


(2) 


where  [A]  is  the  elastic  extensional  matrix,  [D]  is  the  flexural  rigidity  matrix,  and 
[B]  is  the  extension  coupling  matrix  of  the  laminated  plate. 

Element  Displacements,  Matrices  and  Equations 

Proceeding  from  this  point,  the  displacements  in  equation  (1)  are  approximated 
over  a  typical  plate  element  ,  e.g.  rectangular  [8]  or  triangular  [9],  using  the 
corresponding  interpolation  functions.  The  in-plane  displacements  and  the  linear 
strains  are  interpolated  from  nodal  values  by 


1376 


where  [Hm]  and  [Bm]  denote  the  displacement  and  strain  interpolation  matrices, 
respectively,  and  {Wm}  is  the  in-plane  nodal  displacement  vector.  The  transverse 
displacement,  slopes  and  curvatures  are  interpolated  from  the  nodal  values  by 

w  =  [Hb]{w^},  |^’^|  =  [G]{wb},  {K}  =  [B|,]{wb}  (4a,b,c) 


where  [HJ  and  [G]  and  [BJ  denote  the  bending  displacement,  slope  and  curvature 
interpolation  matrices,  respectively,  and  {wb  }  denotes  the  nodal  transverse 
displacements  and  its  derivatives.  Through  the  use  of  Hamilton’s  principle,  the 
equations  of  motion  for  a  plate  element  undergoing  large  amplitude  vibration  may 
be  written  in  the  form 


[hb] 

0 


T  [H] 

[kB]l 

r[klNm]+[klNB]  [klbml 

,r[k2b]  < 

[k»]J 

L  ^'"*1  0  J' 

[  0  ( 

or 

[m]{  w}  +  m  +  [kl]  +  [k2]  }{w}  =  0  (5) 


where  [m]  and  [k]  are  constant  matrices  representing  the  element  mass  and  linear 
stiffiaess  characteristics,  respectively;  Pel]  and  Pe2]  are  the  first  order  and  second 
order  non-linear  sfiffiiess  matrices,  respectively;  PcInJ  depends  linearly  on 
unknown  membrane  displacement  ({Nm  }=  [A][Bni]{wni});  PcInb]  depends  linearly 
on  the  unknown  transverse  displacement  ({NB}=[B][Bb]{Wb});  DelbnJ  depends 
linearly  on  the  unknown  plate  slopes  and  represents  coupling  between  membrane 
and  bending  displacements;  and  [k2b]  depends  quadratically  on  the  unknown  plate 
slopes. 


System  Equations 

After  assembling  the  individual  finite  elements  for  the  complete  plate  and 
applying  the  kinematic  boundary  conditions,  the  finite  element  system  equations  of 
motion  for  the  large-amplitude  free  vibration  of  a  thin  laminated  composite  plate 
can  be  expressed  as 

[M]{w}4-  ([K]+  [K1(W)]  +  [K2(W)]){W}  =  0  (6) 

where  [M]  and  [K]  are  constant  matrices  and  represent  the  system  mass  and 
stifiBness  respectively;  and  [Kl]  and  [K2]  are  the  first  and  second  order  nonlinear 
stifi&iess  matrices  and  depend  linearly  and  quadratically  on  the  unknown  structural 


1377 


nodal  displacements  { W},  respectively.  Most  of  the  finite  element  large  amplitude 
fi-ee  vibration  results  for  plates  and  beams  in  the  literature,  e.g.  references  [1,6]  and 
others,  were  based  on  eq.  (6)  using  an  iterative  scheme  and  various  approximate 
procedures.  The  system  equations  are  not  suitable  for  direct  numerical  integration 
because:  a)  the  nonlinear  stif&iess  matrices  [Kl]  and  [K2]  are  functions  of  the 
unknown  nodal  displacements,  and  b)  the  number  of  degrees  of  fireedom  (DOF)  of 
the  system  nodal  displacements  {W}  is  usually  too  large.  Therefore,  eq.  (6)  has  to 
be  transformed  into  modal  or  generalized  coordinates  followed  by  a  reduction  of 
the  number  of  DOF.  In  addition,  the  general  DufiSng-type  modal  equations  will 
have  constant  nonlinear  modal  stifihiess  matrices.  This  is  accomplished  by  a  modal 
transformation  and  truncation 


r=l 

where  and  cou  are  the  natural  mode  (normalized  with  the  maximum 

component  to  unity)  and  linear  firequency  from  the  eigen-solution  ©|_j.[M]{ 
(|))«=[K]{  (|)}®. 

The  nonlinear  stififtiess  matrices  [Kl]  and  [K2]  in  eq.  (6)  can  now  be  expressed 
as  the  sum  of  the  products  of  modal  coordinates  and  nonlinear  modal  stiffaess 
matrices  as 


[Kl]  =  ^q,(t)[Kl((])«)]  (8) 

r=I 

n  n 

[K2]  =  ^  ^  qr  (t)qs  (t)[K2((|)('') ,  )]  (9) 

r=l  s=l 

The  nonlinear  modal  stifihiess  matrices  [Kl]^"^  and  [O]^'^^  are  assembled  from  the 
element  nonlinear  modal  stif&iess  terms  [kl]^^^  and  Pc2]^“^  as 

([K1]«.[K2]‘“))=  ^([klf>,[k2]'“>)  (10) 

al!  elements 
+  bdy.  conds. 


where  the  element  nonlinear  modal  stif&iess  matrices  are  evaluated  with  the  known 
linear  mode  Thus,  the  nonlinear  modal  stiffiiess  [Kl]^^^  and  [K2]^'®^  are  constant 
matrices.  Equation  (6)  is  thus  transformed  to  the  general  Duf6ng-type  modal 
equations  as 


1378 


(11) 


[M]{q}+([K]+[KlJ+[K2„]){q}  =  0 
where  the  modal  mass  and  linear  stifhiess  matrices  are  diagonal 
([M],[K])  =  [<I.f([M],[K])[<I.] 
and  the  quadratic  and  cubic  terms  are 


LI 

[Kl,]{q}  =  [<Df  J^qjKl] 


1^) 


Vt=1 
n  n 


(rs) 


r=:l  s=l 


\Wk} 


(12) 


(13) 


(14) 


AH  modal  matrices  in  eq.  (11)  are  constant  matrices.  With  given  initial 
conditions,  the  modal  coordinate  responses  {q}  can  be  determined  from  eq.  (11) 
using  any  direct  numerical  integration  scheme  such  as  the  Runge-Kutta  or 
Newmark-P  method.  Therefore,  no  updating  of  the  vibration  modes  is  needed  [10]. 
For  periodic  plate  osciHations  have  the  same  period  T,  the  response  of  all  modal 
coordinates  should  also  have  the  same  period  T.  Since  the  initial  conditions  wHl 
affect  greatly  the  modal  response,  the  determination  of  initial  conditions  for  periodic 
plate  osciHations  is  to  relate  each  of  the  rest  modal  coordinates  in  powers  of  the 
dominated  coordinate  as 


arqi(t;IC)  +  brqi(t;IC)  +  Crq^(t;IC)+ . =  qr(t;IC),  r  =  2,3.  ...n 

(15) 

where  the  2k,  br,  Cr  ,  .  are  constants  to  be  determined,  and  IC  denotes  initial 

conditions.  For  a  three-mode  (n=3)  system,  it  is  accurate  enough  to  keep  up  to  the 
cubic  term  only  in  eq.  (15)  and  this  leads  to  two  set  of  equations 

a2qi(tp;A,B,C)  +  b2qf(tp;A,B,C)  +  C2qJ(tp;A,B,C)  =  q2(tp;A,B,C),  p  =  1,2,3 
a3qi  (tp ; A,  B, C)  +  bsq J  (t p ;  A,  B.  C)  +  Csq J  (tp ; A.  B,  C)  =  q3  (tp ; A,  B,  C),  p  =  1,2,3 

(16a,b) 

in  which  the  modal  coordinates  qi,  q2  and  qs  at  tp  are  known  quantities  and  the 
initial  conditions  are  qi(0)=A,  q2(0)  =B,  q3(0)=C  and  qi(0)  =  q2(0)  =  q3(0)  =  0. 
PracticaHy,  only  eight  equations  are  needed  to  determine  the  eight  unknowns  a2,  as. 


1379 


hi,  b3,  C2,  C3,  B  and  C  through  an  iterative  scheme.  However,  the  number  of 
equations  can  be  more  than  the  number  of  unknowns  for  accurate  determination  of 
initial  conditions  and  the  least  square  method  is  employed  in  this  case. 

The  time  history  of  the  plate  maximum  deflection  can  be  obtained  from  eq.  (7). 
The  participation  value  from  the  r  th  linear  mode  to  the  total  deflection  is  defined 
as 


maxjqj. 

n 

^max|qi 

i=l 


(17) 


Thus,  the  minimum  number  of  the  linear  modes  for  an  accurate  and  converged 
frequency  solution  can  be  determined  based  on  the  modal  participation  values. 


Results  and  Discussions 

Assessment  of  Single-Mode  Elliptic  Function  Solution 

The  fundamental  frequency  ratio  Co/col  =  2.3051  at  Wmax/r  =5.0  for  a  simply 
supported  beam  obtained  by  Woinowsky-Kiieger  [2]  using  a  single-mode  and 
elliptic  fimetion  is  assessed  first.  The  conventional  beam  element  having  six  (four 
bending  and  two  axial)  DOF  is  used.  A  half-beam  is  modeled  with  10,  15,  20 
elements,  and  the  lowest  four  symmetrical  linear  modes  are  used  in  the  Duffing 
modal  equations.  Table  1  shows  that  a  20-element  and  1-mode  model  will  yield  a 
converged  result.  The  percentages  of  participation  from  each  mode  for  various 
values  of  Wmax/r  are  given  in  Table  2.  The  modal  participation  values  demonstrate 
that  a  single  mode  (n=l)  will  yield  an  accurate  fundamental  frequency  because  the 
contribution  from  higher  linear  modes  to  the  total  deflection  is  negligible  (<  0.01  % 
for  Wmax/r  up  to  5.0).  There  is  a  small  difference  in  frequency  ratios  between  the 
present  finite  element  and  the  elliptic  integral  solutions.  This  is  due  to  the  difference 
between  the  axial  forces  of  the  two  approaches,  the  finite  element  method  (FEM) 
gives  a  non-uniform  axial  force  in  each  element;  however,  the  average  value  of  the 
axial  force  for  each  element  is  the  same  as  the  one  in  the  classic  continuum 
approach.  The  lowest  (2.0310)  and  the  highest  (3.3438)  frequency  ratios  at  Wmax/r 
=5.0  in  reference  [5]  are  not  accurate. 

Frequency  ratios  for  higher  modes  of  the  simply  supported  beam  are  obtained 
next.  A  model  with  40-elements  and  3-anti-symmetric  modes  for  the  whole  beam  is 
employed  for  the  frequency  ratio  of  the  second  nonlinear  mode.  The  mode 
participations  shown  in  Table  2  indicate  that  a  single-mode  approach  will  yield 
accurate  frequency  results.  And  the  frequency  ratios  for  the  second  mode  are  the 
same  as  those  of  the  fundamental  one.  Thus,  the  present  method  agrees  extremely 
well  with  Woinowsky-Krieger’s  classic  single-mode  approach. 


1380 


The  time  history  of  the  first  two  symmetric  modal  coordinates  and  the  beam 
central  displacement,  phase  plot,  and  power  spectral  density  (PSD)  at  maximum 
beam  deflection  W^Jr  =  5.0  for  the  fimdamental  firequency  (or  mode)  are  shown  in 
Fig.  1.  The  time  scale  is  non-dimensional  and  Ti  is  the  period  of  the  fundamental 
linear  resonance.  It  is  noted  that  although  the  central  displacement  response  looks 
like  a  simple  harmonic  motion,  it  does  have  a  small  deviation  fi:om  pure  harmonic 
motion  due  to  the  second  small  peak  in  the  spectrum.  This  is  in  agreement  with 
classical  solution  that  the  ratio  of  the  jfrequency  of  the  second  small  peak  to  that  of 
the  first  dominant  peak  is  3. 

Now  we  are  ready  to  assess  the  single-mode  fundamental  firequency  of  a  simply 
supported  square  plate  obtained  by  Chu  and  Herrmann  [3].  A  quarter  of  the  plate  is 
modeled  with  6  x  6,  7  x  7,  8  x  8  and  9  x  9  mesh  sizes  and  1,  2,  4  or  5  symmetrical 
modes  are  used.  The  conforming  rectangular  plate  element  with  24  (16  bending 
and  8  membrane )  DOF  is  used.  The  in-plane  boundary  conditions  are  u  =  v  =  0  on 
all  four  edges.  Table  3  shows  that  the  8  x  8  mesh  size  in  a  quarter-plate  and  4- 
mode  model  should  be  used  for  a  converged  and  accurate  frequency  solution.  Table 
4  shows  the  fi:equency  ratios  and  modal  participation  values  for  the  lowest  three 
modes  at  various  Wmax/h  for  a  simply  supported  square  plate  (8x8  mesh  size  in  a 
quarter-plate).  It  indicated  that  at  least  two  linear  modes  are  needed  for  an  accurate 
frequency  prediction  at  Wmax/h  =1.0,  and  the  contribution  of  higher  linear  modes 
increase  with  the  increase  of  plate  deflections.  The  modal  participation  values  also 
show  that  the  combined  modes  (1,3)-(3,1)  and  (2,4)-(4,2)  are  independent  of  the 
large-amplitude  vibrations  dominated  by  (1,1)  and  (2,2)  modes,  respectively.  The 
time  history,  phase  plot,  and  PSD  at  the  maximum  deflection  Wmax/h  =1.0  for  the 
fundamental  mode  are  shown  in  Fig.  2a,  and  Tu  is  the  period  of  the  fundamental 
linear  resonance.  There  is  one  small  peak  in  the  spectrum  and  the  frequency  ratio  of 
the  second  small  peak  to  the  first  dominant  one  is  3.  The  low  (1.2967)  and  the  high 
(1.5314)  frequency  ratios  at  Wmax/h  =1.0  given  in  reference  [6]  are  not  accurate. 

The  influence  of  the  initial  conditions  on  periodic  motion  is  demonstrated  in  Fig. 
2a  and  2b.  In  Fig.  2a,  the  modal  coordinates  all  have  the  same  period,  and  the  initial 
conditions  are  determined  fi*om  eq.(15).  They  are  qii(0)/h=1.0,  qi3+3i(0)/h=  - 
0.0155,  qi3.3i  (0)/h  =  0.0,  q33(0)/h=0.000813,  and  qi5+5i(0)/h=  0.00011,  and  initial 
velocities  are  null,  whereas  in  Fig.  2b,  qii(0)/h=1.0  and  all  others  are  nuU.  The 
modal  coordinates  do  not  have  the  same  period. 

Clamped  Beam 

It  is  thus  curious  to  find  out  whether  multiple-mode  is  required  for  the  clamped 
beam.  Convergence  study  of  the  fundamental  firequency  ratios  at  Wmax/r  =3.0  and 
5.0  shown  in  Table  5  indicates  that  a  25-element  (half-beam)  and  4-mode  model 
win  yield  accurate  and  converged  results.  The  time  history,  phase  plot  and  PSD  at 
Wmax/r  =5.0  are  shown  in  Fig.  3.  The  modal  participation  values  in  Table  6  and  the 
PSD  in  Fig.  3  confirm  that  at  least  two  modes  are  needed  for  accurate  firequency 
results. 


1381 


Symmetric  Composite  Plate 

A  simply  supported  eight-layer  symmetrically  laminated  (0/45/-45/90)s 
composite  plate  with  an  aspect  ratio  of  2  is  investigated.  The  graphite/epoxy 
material  properties  are  as  follows;  Young’s  moduli  Ei  =  155  GPa,  E2  =  8.07  GPa, 
shear  modulus  Gn  =  4.55GPa,  Poisson’s  ratio  V12  =0.22,  and  mass  density  p  = 
1550  kg/m^  A  12  X  12  mesh  is  used  to  model  the  plate.  The  in-plane  boundary 
conditions  are  fixed  (u=v=0)  at  all  four  edges.  The  first  seven  linear  modes  are  used 
as  the  modal  coordinates.  Table  7  gives  the  fundamental  firequency  ratios  and  mode 
participation  values  for  the  linear  modes  in  increasing  firequency  order.  The  modal 
participation  values  indicate  clearly  that  four  modes  are  needed  in  predicting  the 
nonlinear  fi-equency,  and  other  three  of  the  seven  are  independent  of  the 
fundamental  nonlinear  mode.  Figure  4  shows  the  time-history,  phase  plot,  and  PSD 
at  Wmax/h  =1.0. 


UNS  YMMETRIC  COMPOSITE  PLATE 

A  simply  supported  two-layer  laminated  (0/90)  composite  plate  of  15  x  12  x 
0.048  in.  (38  x  30  x  0.12  cm)  is  investigated.  The  graphite/epoxy  material 
properties  are  the  same  as  those  of  the  symmetric  composite  plate.  A 12  x  12  mesh 
is  used  to  model  the  plate.  The  in-plane  boundary  conditions  are  fixed  at  all  four 
edges.  The  first  four  linear  modes  are  used  as  the  modal  coordinates.  Table  8  gives 
the  fundamental  firequency  ratios  and  mode  participation  values  for  the  linear  modes 
in  increasing  fi-equency  order.  From  the  phase  plot,  the  time  histories  and  PSD 
shown  in  Fig.  5,  it  can  be  seen  that  the  total  displacement  response  has  a  non-zero 
mean  (i.e.  the  positive  and  negative  displacement  amplitudes  for  all  modal 
coordinates  are  not  equal).  The  quasi-ellipse  in  the  phase  plot  is  not  symmetrical 
about  the  vertical  velocity-axis.  In  the  PSD  at  Wmax/h  =1.0,  it  is  observed  that  there 
are  four  small  peaks  in  the  spectrum  and  the  firequency  ratios  of  the  second,  third, 
fourth  and  fifth  peak  to  the  first  dominant  one  are  2,  3,  4  and  5,  respectively.  This 
observation  indicates  that  the  displacement  response  includes  the 
superharmonances  of  orders  2,  3,  4,  and  5.  The  curves,  which  the  positive  and 
negative  displacement  amplitudes  are  plotted  against  the  fundamental  firequency 
ratio,  are  also  given  in  Fig.  5.  The  difference  between  the  positive  and  negative 
amplitudes  increases  as  the  firequency  ratio  increasing. 


Conclusions 

A  multimode  time-domain  formulation,  based  on  the  finite  element  method,  is 
presented  for  nonlinear  firee  vibration  of  composite  plates.  The  use  of  FEM  enables 
the  present  formulation  to  deal  with  composite  plates  of  complex  geometries  and 
boundary  conditions,  and  the  use  of  the  modal  coordinate  transformation  enables  to 
reduce  the  number  of  ordinary  nonlinear  differential  modal  equations  to  a  much 
smaller  one.  The  present  procedure  is  able  to  obtain  the  general  Duffing-type  modal 
equations  easily.  Initial  conditions  for  all  modal  coordinates  having  the  same  time 


1382 


period  are  presented.  The  participation  value  of  the  linear  mode  to  the  nonlinear 
deflection  is  quantified  ;  they  can  clearly  determine  the  minimum  number  of  linear 
modes  needed  for  accurate  nonlinear  frequency  results. 

The  present  fundamental  nonlinear  fi-equency  ratios  have  been  compared  with 
the  single-mode  solution  obtained  by  Woinowsky-Kneger  for  simply  supported 
beams  and  by  Chu  and  Herrmann  for  simply  supported  square  plates.  The 
Woinowsky-Krieger’s  single-mode  solution  is  accurate.  For  all  other  solutions, 
however,  two  or  more  modes  are  needed.  The  nonlinear  firequencies  for 
symmetrically  and  unsymmetrically  laminated  rectangular  composite  plates  are  also 
obtained.  The  phase  plot  and  power  spectral  density  showed  that  nonlinear 
displacement  responses  are  no  longer  harmonic,  and  multiple  modes  are  required 
for  isotropic  clamped  beams  and  isotropic  and  composite  plates. 


References 

1.  M.  Sathyamoorthy  1987  Applied  Mechanics  Review  40,  1553-1561.  Nonlinear 
vibration  analysis  of  plates:  A  review  and  survey  of  current  developments. 

2.  S.  Woinowsky-Kreger  1950  Journal  of  Applied  Mechanics  17,  35-36.  The  effect  of  an 
axial  force  on  the  vibration  of  hinged  bars. 

3.  H.  N.  Chu  and  G.  Herrmann  1956  Journal  of  Applied  Mechanics  23,  523-540. 
Influence  of  large  amplitudes  on  jfree  flexural  vibrations  of  rectangular  elastic  plates. 

4.  G.  Singh,  A.  k.  Sharma  and  G.  V.  Rao  1990  Journal  of  Sound  and  Vibration  142,  77- 
85.  Large  amplitude  free  vibration  of  beams-discussion  of  various  formulations  and 
assumptions. 

5.  H.  R.  Srirangaraja  1994  Journal  of  Sound  and  Vibration  175,  425-427.  Nonlinear  free 
vibrations  of  uniform  beams. 

6.  S.  R.  Rao,  A.  H.  Sheikh  and  M.  Mukhopadhyay  1993  Journal  of  the  Acoustical 
Society  of  America  93  (6),  3250-3257.  Large-amplitude  finite  element  flexural 
vibration  of  plates/stiffened  plates. 

7.  Y.  Shi  and  C.  Mei  1996  Journal  Sound  and  Vibration  193,  453-464.  A  finite  element 
time  domain  modal  formulation  for  large  amplitude  free  vibration  of  beams  and  plates. 

8.  K.  Bogna:,  R.  L.  Fox  and  L.  A.  Schmit  1966  Proceeding  of  Conference  on  Matrix 
Methods  in  Structural  Mechanics,  AFFDL-TR-66-80,  Wright-Patterson  Air  Force 
Base,  Ohio,  October  1965,  397-444.  The  gena-ation  of  interelement  compatible 
stiffness  and  mass  matrix  using  the  interpolation  formulas. 

9.  Teseller  and  T.  J.  R.  Hughes  1985  Computer  Methods  in  Applied  Mechanics  and 
Engineering  50,  71  -101 .  A  three  node  Mindlin  plate  element  with  improved  fransverse 
shear. 

10.  A.  K.  Noor  1981  Composites  and  Structures,  13,  31-44.  Recent  advances  in  reduction 
methods  for  nonlinear  problems. 


1383 


Table  1 .  Convergence  of  the  fundamental  frequency  ratio  at  Wmax/r  =5.0  for  a 
simply  supported  beam 


No.  of  elements 
and  4  modes 

(C0/(0l)i 

No.  of  modes 
and  20  elements 

(G)/G)l)i 

10 

2.3537 

1 

2.3506 

15 

2.3511 

2 

2.3506 

20 

2.3506 

3 

2.3506 

-- 

4 

2.3506 

Table  2  The  lowest  two  frequency  ratios  and  the  modal  participations  for  a 
simply  supported  beam 


W^nax/r 

Elliptic 
integral  [2] 
(CO/OOl)! 

FEM 

Modal 

Participation  % 

(CO/COl)! 

_ _ 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0331 

100.00 

0.000 

0.000 

0.8 

1.0580 

1.0581 

100.00 

0.000 

0.000 

1 

1.0892 

1.0892 

100.00 

0.000 

0.000 

2 

1.3178 

1.3179 

100.00 

0.002 

0.000 

3 

1.6257 

1.6258 

100.00 

0.004 

0.000 

4 

1.9760 

1.9761 

99.99 

0.005 

0.000 

5 

2.3501 

2.3506 

99.99 

0.009 

0.000 

W.„ax/r 

(co/g)l)2 

(C0/CDl)2 

q.2_  _ 

q4 

Q6 

0.2 

1.0038 

1.0038 

100.00 

0.000 

0.000 

0.4 

1.0150 

1.0149 

100.00 

0.000 

0.000 

0.6 

1.0331 

1.0332 

100.00 

0.000 

0.001 

0.8 

1.0580 

1.0582 

100.00 

0.000 

0.001 

1 

1.0892 

1.0893 

100.00 

0.000 

0.002 

2 

1.3178 

1.3181 

99.99 

0.000 

0.006 

3 

1.6257 

1.6260 

99.98 

0.000 

0.015 

4 

1.9760 

1.9768 

99.98 

0.000 

0.021 

5 

2.3501 

2.3512 

99.96 

0.000 

0.037 

Table  3.  Convergence  of  the  fundamental  frequency  ratios  for  a  simply 
supported  square  plate  (Poisson’s  ratio=0.3) 


Mesh  sizes 
and 

(CD/Qjn 

at 

W^,Jh 

No.  of  modes 
and 

(co/coUn 

at 

W^Jh 

4  modes 

1.0 

1.4 

8x8  mesh 

1.0 

1.4 

6x6 

1.4174 

1.7423 

1 

1.7028 

7x7 

1.4163 

1.7396 

2 

1.4169 

1,7433 

8x8 

1.4164 

1.7403 

4 

1.4164 

1.7403 

9x9 

1.4164 

1.7400 

5 

1.4163 

1.7401 

Table  4.  The  lowest  three  frequency  ratios  and  the  modal  participations  for  a 
simply  supported  square  plate  (Poisson’s  ratio=0.3) 


Elliptic 

FEM 

integral [3] 

(CO/ffiiJii 

(to/oiOii 

Modal 

Participation 

% 

qu 

qi3  -1-  qsi 

qi3  -  qsi 

q33 

qi5  -f.  qsi 

0.2 

1.0195 

1.0195 

99.93 

0.07 

0.00 

0.00 

0.00 

0.4 

1.0757 

1.0765 

99.72 

0.27 

0.00 

0.01 

0.00 

0.6 

1.1625 

1.1658 

99.38 

0.59 

0.00 

0.02 

0.00 

0.8 

1.2734 

1.2796 

98.93 

1.02 

0.00 

0.05 

0.01 

1.0 

1.4024 

1.4163 

98.34 

1.57 

0.00 

0.08 

0.01 

1.2 

1.5448 

1.5659 

97.54 

2.30 

0.00 

0.15 

0.01 

1.4 

1,6933 

1.7401 

96.29 

3.42 

0.00 

0.27 

0.02 

(C0/®l)21 

q2i 

q23 

q4i 

q43 

... 

0.2 

N/A 

1.0243 

99.93 

0.06 

0.01 

0.00 

... 

0.4 

N/A 

1.0976 

99.50 

0.45 

0.03 

0.01 

— 

0.6 

N/A 

1.2072 

98.15 

1.28 

0.54 

0.02 

... 

0.8 

N/A 

1.3411 

97.54 

2.41 

0.00 

0.05 

— 

1.0 

N/A 

1.5126 

96.24 

3.69 

0.00 

0.08 

... 

1.2 

N/A 

1.6900 

94.90 

4.92 

0.03 

0.15 

... 

1.4 

N/A 

1.8952 

93.54 

6.01 

0.01 

0.44 

... 

y/raJh 

(C0/0)l)22 

q22 

q24  +  q42 

q24  -  q42 

q44 

... 

0.2 

N/A 

1.0245 

100.00 

0.00 

0.00 

... 

0.4 

N/A 

1.0751 

100.00 

0.00 

0.00 

0.00 

... 

0.6 

N/A 

1.1611 

99.99 

0.00 

0.00 

0.01 

... 

0.8 

N/A 

1.2806 

99.98 

0.01 

0.00 

0.01 

... 

1.0 

N/A 

1.4041 

99.93 

0.01 

0.00 

0.06 

— 

1.2 

N/A 

1.5551 

99.97 

0.01 

0.00 

0.01 

... 

1.4 

N/A 

1.7074 

99.98 

0.02 

0.00 

0.00 

— 

Table  5,  Convergence  of  the  fundamental  frequency  ratios  for  a  clamped  beam 

No.  of 
elements 

and  4  modes 

(co/©l)i 

at 

W™/r 

No.  of  modes 
and 

(©/©lOi 

at 

Wn.ax/r 

3.0 

5.0 

25  elements 

3.0 

5.0 

10 

1.1751 

1.4046 

1 

15 

1.1740 

1.4009 

2 

20 

1.1732 

1.3999 

3 

25 

1.1731 

1.3996 

4 

1385 


Table  6  The  fundamental  frequency  ratios  and  the  modal  participations  for  a 
clamped  beam 


Elliptic 

integral 

(0)/(0l)i 

FEM 

(ffl/(0L)l 

Modal  Participation 

% 

qi 

q3 

qs 

_ 

1.0222 

1.0222 

99.78 

0.20 

0.02 

0.00 

2.0 

1.0857 

1.0841 

99.33 

0.58 

0.08 

0.02 

3.0 

1.1831 

1.1731 

98.35 

1.44 

0.17 

0.04 

4.0 

1.3064 

1.2817 

97.37 

2.28 

0.29 

0.07 

5.0 

1.4488 

1.3996 

96.26 

3.22 

0.42 

0.11 

Table  7  The  fundamental  frequency  ratios  and  the  modal  participations  for  a 

simply  supported  rectangular  (0/45/-45/90)s  composite  plate  (a/b=2) 


WmJh 

Modal 

Participation 

% 

(ffl/coOn 

qu 

qi2 

q2i 

qi3 

q22 

q23 

qsi 

0.2 

1.0408 

99.51 

0.00 

0.00 

0.41 

0.07 

0.00 

0.02 

0.4 

1.1490 

96.57 

0.00 

0.00 

3.01 

0.24 

0.00 

0.17 

0.6 

1.3484 

92.93 

0.00 

0.00 

4.55 

0.47 

0.00 

2.04 

0.8 

1.5241 

98.51 

0.00 

0.00 

0.53 

0.94 

0.00 

0.02 

1.0 

1.7190 

97.43 

0.00 

0.00 

2.39 

0.11 

0.00 

0.07 

1.2 

1.9258 

95.78 

0.00 

0.00 

3.57 

0.62 

0.00 

0.02 

1.4 

2.1409 

94.27 

0.00 

0.00 

4.84 

0.77 

0.00 

0.13 

Table  8 

The  fundamental  frequency  ratios  and  the  modal  participations  for  a 
simply  supported  rectangular  (0/90)  composite  plate 

W„,Jh 

Modal  Participation 

% 

(CO/COiOn 

qii 

qi3 

qsi 

q33 

0.2 

1.0358 

97.83 

1.18 

0.82 

0.18 

0.4 

1.1432 

95.13 

2.25 

2.24 

0.38 

0.6 

1.2993 

94.53 

3.86 

1.18 

0.60 

0.8 

1.5432 

88.56 

4.36 

4.77 

2.31 

1.0 

1.7880 

89.15 

3.31 

5.06 

2.48 

1.2 

2.0142 

92.22 

2.89 

3.15 

1.74 

1.4 

2.2823 

92.01 

2.92 

2.92 

2.15 

1386 


0.5  1  1.5  2  -6  -3 

Time  Ratio  (tn‘1) 


4  6  8  10  12 
Frequency  Ratio 


U 

Displacement 


Figure  1.  Time  histories,  phase  plot  and  PSD  for  the  fundamental 
mode  at  =5.0  of  a  simply  supported  beam 


Power  Spectrum  Density  Total  OlsplacemonllThIckness  qSSn'hJcknBSS  q1 1/ThIckness 

p  P  P  T*  Thousandths  o  o  o 


Time  Ratio  (tn"11) 


-0.4  -0.2  0  0.2  0.4 

Displacement 


Figure  2a.  Time  histories,  phase  plot  and  PSD  for  the  fundamental 
mode  at  W^u/h  =1.0  of  a  simply  supported  square  plate 


1388 


Thousandths 


Displacement  Spectrum  Density  Total  Displacement/Thickness  q22/Thickness  q1  IH'hickrtess 


0  12  3  4 

Time  Ratio  {t/T11) 


0  12  3  4 


0  2  4  6  8  10  12 

Frequency  Ratio 


0  12  3  4 


Time  Ratio  {tfTII) 


0  12  3  4 

Time  Ratio  (t/TII) 


Displacement 


Figure  4.  Time  histories,  plot  and  PSD  for  the  fundamental 
mode  at  Wmax/h  =1.0  of  a  simply  supported 
(0/45M5/90)s  rectangular  plate 


1391 


Power  Specirum  DonsUy  Total  Dlsplacemenin'hlckness  qSin’hlckness  q11/Thlckness 


NONLINEAR  FORCED  VIBRATION  OF  BEAMS  BY  THE 
HIERARCHICAL  FINITE  ELEMENT  METHOD 

P.  Ribeiro  and  M.  Petyt 

Institute  of  Sound  and  Vibration  Research,  University  of  Southampton,  Southampton 

S017  IBJ,  UK 

Abstract:  The  hierarchical  finite  element  (HFEM)  and  harmonic  balance 
methods  are  used  to  derive  the  equations  of  motion  of  beams,  in  steady-state  forced 
vibration  with  large  amplitude  displacements.  These  equations  are  solved  by  the 
Newton  and  continuation  methods.  The  stability  of  the  obtained  solutions  is 
investigated  by  studying  the  evolution  of  perturbations  of  the  solutions.  Additionally,  a 
method  that  allows  a  quick  examination  of  the  stability  of  the  solution  is  presented  and 
applied.  The  convergence  properties  of  the  HFEM,  the  influence  of  the  number  of 
degrees  of  freedom  and  of  in-plane  displacements  are  discussed.  The  HFEM  results  are 
compared  with  experimental  results.  Symbolic  computation  is  used  in  the  derivation  of 
the  model. 


NOTATION 


A  -  extension  coefficient 
b  -  width  of  the  beam 
B  -  coupling  coefficient 

[C]  -  damping  matrix 
D  -  bending  coefficient 

[D]  -  Jacobian  of  {F} 

|D|  -  determinant  of  [D] 

E  -  Young's  modulus 

[E]  -  elastic  matrix 

{f}  -  vector  of  out-of  plane  shape  functions 
|F}  -  vec.  of  amplitudes  of  generalised  forces 

-  vector  of  generalised  forces 

{F}  -  vector  of  dynamic  forces 

{g}  -  vector  of  in-plane  shape  functions 

h  -  length  of  the  finite  elements 

h  -  thickness  of  the  beam 

[Klb]  -  linear  bending  stiffiiess  matrix 

[Kip]  -  linear  stretching  stiffiiess  matrix 

[K2],  [K3]  and  [K4]  -  components  of 

nonlinear  stiffiiess  matrix 

[Knl]  -  nonlinear  stiffiiess  matrix 

L  -  length  of  the  beam 

[M]  -  mass  matrix 

[Mb]  -  bending  mass  matrix 
[Mp]  -  in-plane  mass  matrix 

[N]  -  matrix  of  shape  functions 
[N'''(x)J  -  row  matrix  of  out-of-pl.  sh.  f 

p  j  -  number  of  in-plane  shape  functions 


pQ  -  number  of  out-of-plane  s.  funct. 

{qp}  -  in-plane  displacement  fimction 
{qw}  -  transverse  displacement  function 
r  -  radius  of  gyration 
t  -  time 

u  -  in-plane  displacement 
Ur  -  generalised  in-plane  displacements 
w  -  transverse  displacement 
Wr  -  generalised  out-of-pl.  displ. 

{wc},  (wj  -  coef  of  cosine  and  sine  terms 
X  -  axial  coordinate  of  the  beam 

a  -  loss  factor 
p  -  damping  factor 

eg  ,s§  -  linear  membrane  and  bending  strains 
-  geometrically  nonlin.  membrane  strain 
{si },  {£2 }-  linear  and  geom.  nonl.  strains 
5  W  -  virtual  work  of  the  external  forces 

cx 

SWy  -  virtual  work  of  the  internal  forces 
5W-  -  virtual  work  of  the  inertia  forces 

in 

X  -  characteristic  exponent 

p  -  mass  density 

CO  -  angular  fi:equency 

cOjj.  -  natural  frequencies 

Ja>y  -  diagonal  matrix  of  squares  of 

natural  firequencies 
-  Viscous  damping  ratio 


1393 


1- INTRODUCTION 


In  real  systems,  due  to  large  amplitudes  of  the  excitation,  small  stiffiiess  or 
excitation  with  a  frequency  in  the  neighbourhood  of  resonance  frequencies,  vibrations 
with  large  amplitudes  can  occur.  In  this  case,  the  linear  theories  may  not  allow  a  good 
representation  of  the  dynamic  characteristics  of  the  system. 

A  typical  case  study  of  vibrations  in  the  nonlinear  regime  is  the  forced  vibration 
of  beams,  with  large  displacements.  Although  a  large  amount  of  investigation  has  been 
carried  out  in  this  field  [1,  2,  10,  18  and  others],  a  method  that  would  allow  the 
inclusion  of  higher  order  mode  contributions  and  damping,  without  increasing 
excessively  the  number  of  degrees  of  freedom  (d.o.f )  is  desirable.  The  purpose  of  this 
work  is  to  apply  and  investigate  the  advantages  of  a  method  that  satisfies  these 
conditions:  the  hierarchical  finite  element  method  (HFEM). 

In  the  HFEM,  to  achieve  better  approximations,  higher  order  shape  functions 
are  added  to  the  existing  model.  Convergence  tends  to  be  achieved  with  far  fewer 
d.o.f  than  in  the  /z-version  of  the  finite  element  method  [4,  11].  The  linear  matrices 
possess  the  embedding  property,  meaning  that  the  associated  element  matrices  for  a 
number  of  shape  functions  n=ni  are  always  submatrices  for  n=n2,  n2>ni.  The  existing 
nonlinear  matrices  of  an  approximation  of  lower  order,  ni,  can  be  used  in  the  derivation 
of  the  nonlinear,  matrices  of  the  improved  approximation,  n2.  This  makes  the  construction 
of  a  more  accurate  model,  potentially  quicker  in  the  HFEM  than  in  the  /z-version. 

We  are  going  to  consider  that  the  time  variation  of  the  solution  may  be 
expressed  by  harmonics  and  use  the  harmonic  balance  method  (HBM).  Compared  with 
perturbation  methods,  the  main  advantages  of  the  HBM  are  its  simplicity,  the  fact  that 
it  is  not  restricted  to  weakly  non-linear  problems  and,  for  smooth  systems,  the 
assurance  of  convergence  to  the  exact  solution  [5]. 

In  nonlinear  vibrations,  the  frequency  response  curves  can  have  multi-valued 
regions,  turning  and  bifurcation  points.  In  these  regions,  we  are  going  to  use  a 
continuation  method  [8,  14],  because,  if  the  Newton  method  alone  is  applied,  the 
solution  will  depend  heavily  on  the  initial  guess  and  convergence  is  very  difficult  to 
achieve. 

Symbolic  computation  [15]  will  be  utilised,  allowing  an  easier  and  more 
accurate  construction  of  the  model. 


1394 


2  -  MATHEMATICAL  MODEL 


The  beam  is  assumed  to  be  elastic  and  isotropic,  with  thin  uniform  thickness  h.  The 
effects  of  transverse  shear  deformations  and  rotatory  inertia  are  neglected.  The  transverse 
displacement  is  large  compared  with  the  beam  thickness,  but  is  very  small  compared  with 
the  length  of  the  beam  (w  «  L).  The  slopes  are  also  very  small;  (w  « 1 . 


The  displacement  components  u  and  w  may  be  expressed  as  the  combination  of 
the  hierarchical  polynomial  shape  functions, 


§lS2"-Spi  ^ 

0  f,f,..-fn 


{d}  =  |;;)  =  [N]{q}.[N]  = 

{q}"=[U,  ...  U,,  W.  ...  Wj.  (2.1) 

The  hierarchical  polynomials  used  in  this  study  were  derived  from  Rodrigue's  form 
of  Legendre  polynomials  [4].  Only  one  element  was  used  to  model  the  whole  beam  and 
only  the  shape  functions  that  satisfy  the  boundary  conditions  were  included  in  the  model. 
Applying  the  theory  of  Bemoulli-Euler,  expressing  the  strain ; 


1  as 


{s}  = 


(w^f  /2 
0 


P 

_JOo 


'Oj  ■ 


(2.2) 

I  0  j  L8“j  [oJ  --  -- 

and  equating  the  virtual  work  of  the  inertia  forces  (D’  Alembert  principle)  to  the 
virtual  work  of  the  external  and  elastic  restoring  forces  we  obtain: 

{5q}^{F}-bJJ{5s,}%{5s,r)[E]({e,}  +  {s,})dL=  {5q}^[M]{q},  (2.3) 

°  ;  A.  D  =  £(1, 2^)Ed2 ;  [M]  =  phbJjN]^[N]dL . 


[E]  = 


The  stiffness  matrices  are  defined  by: 


b[{58,}^[E]{s,}dL={5q}"[Kl]{q}.bJj6eT[E]{8,}dL={6qnK2]{q}. 
b({5e3r[E]{8,}dL={6q}"[K3]{q},  b[{5e,}"[E]{84dL={5qnK4]{q} .  (2.4) 

Considering  only  transverse  forces,  if  P(t)  represents  a  concentrated  force  acting 
at  the  point  x=Xj  and  (x,  t)  represents  a  distributed  force,  the  generalised  forces  are 
{F}  =  (£p,(x.t){N”(x)}dL  +  P,(t){N”(x.)}).  (2.5) 

In  real  systems  energy  is  dissipated;  consequently,  damping  should  be  included 
in  the  present  model.  For  a  large  variety  of  materials  experimental  investigations  show 


1395 


that  the  energy  dissipated  per  cycle  is  not  dependent  on  the  frequency  and  is 
proportional  to  the  square  of  the  amplitude  of  vibration  [12,  13],  The  corresponding 
type  of  damping  is  called  hysteretic.  We  will  represent  it  by  a  matrix  proportional  to 
the  mass  matrix  and  frequency  dependent: 

[C]  =  i[M]  (2.6) 

Considering  that  damping  in  the  beam  results  only  from  the  action  of  the  linear  axial 
and  bending  strains,  we  have  the  following  equations  of  motion; 


"0 

_K3 

K4jj 

(2.7) 


The  in-plane  inertia  can  be  neglected  for  slender  beams  [3]  and  the  damping 
contribution  due  to  the  axial  stress  is  generally  negligible  compared  to  that  due  to  the 
bending  stress  [12],  With  these  approximations  and  because  [K3]  =  2[K2]^,  ref  [4],  we 

can  simplify  the  equations  of  motion  to  obtain: 

+ J[M.]{qw}  +  [Kl.]{q„}  +  [Knl]{q„}  =  {F},  (2.8) 

[Knl]  =  [K4]  -  2[K2]^[k1,]'‘[K2]  .  (2.9) 

To  integrate  exactly  terms  involving  shape  functions  or  its  derivatives,  present 
in  the  stiffriess  and  mass  matrices,  symbolic  computation  was  employed,  using  the 
package  Maple  [15]. 

If  the  external  excitation  is  harmonic  and  if  initial  conditions  are  such  that  no 
transient  response  exists,  then  {qw(t)}  may  be  expressed,  in  a  first  approximation,  as: 


{qw(t)}={wc}cos(©t)+{Ws}sin(a)t)  (2.10) 

We  are  going  to  insert  this  equation  into  the  equations  of  motion  (2.8)  and 
apply  the  HBM.  This  method  can  be  easily  implemented  in  a  program  written  with  the 
symbolic  manipulator  For  that,  one  defines  the  command  trign  using  Maple 

library  of  trigonometric  functions,  trig,  in  the  following  way; 
trign:  =readlih( 'trig/reduce')  [15,  17].  trign,  thus  defined,  replaces  all  nonlinear 
trigonometric  functions  by  linear  ones^.  With  the  command  coeff  one  selects  the  terms 
in  cos(ot)  and  sin(a3t).  In  this  way,  we  obtain  equations  of  motion  of  the  form: 


1  For  example  cos^(Q)t) 


is  replaced  by  ^cos(a)t) 


+  .icos(3cot) 


1396 


-CO' 


M,  0 
0  M, 


-3M, 


KL 


(2.11) 


{Fj=|0Knl]{q4cos(cDt)dt=gKM.l  +  iKM.3){wJ+iK^^^  (2.12) 

{F,}=||;[KM]{q4smMdt  =iKNL2{wJ  +  gKNLl  +  ^K^o]{w^  (2.13) 


where  KNLl  is  a  iunction  of  { Wc}  only,  KNL2  is  a  function  ofboth  (wc)  and  (wj  and  ICNL3 
is  a  function  of  {Ws}  only  These  three  matrices  are,  as  well  as  Mb  and  Klb,  symmetric. 


3  -  STUDY  OF  THE  STABILITY  OF  THE  SOLUTIONS 

We  will  study  the  problem  of  local  stability  of  the  harmonic  solution  by  adding 
a  small  disturbance  to  the  steady  state  solution 

{q}  =  {q„}  +  {8qw}  (3.1) 

and  studying  how  the  variation  of  the  solution  evolves.  If  {6q^}  dies  out  with  time 
then  {q^}  is  stable,  if  it  grows  then  {q^}  is  unstable. 

Inserting  the  disturbed  solution  (3.1)  into  equation  (2.8),  expanding  the 
nonlinear  terms  into  Taylor  series  around  {q„}  and  ignoring  terms  of  order  higher 

than  {bq^j ,  we  obtain  the  variational  equation: 

K]{5q„}  +  ^K]{6q„}  +  [KlJ{5q4  +  ^Mj^{5q.}=M  (3-2) 

The  coefficients  are  periodic  functions  of  time.  With  symbolic 

manipulation,  they  can  easily  be  expanded  in  a  Fourier  series.  If  {q^}  is  of  the  form 

(2.10)  and  since  is  quadratic  in  {q„},  we  have: 

3{qw} 

^(M|^=[[p.]  +  [p,]cos(2cst)  +  [p,]sin(2o)t)] .  (3.3) 

2  With  this  formulation,  KNL2  must  be  calculated  using  2[N*  J{wJ[N*J(w,}  ,  otherwise  .iKNL2 
should  be  considered  instead  of  iKNL2  ■ 


1397 


Simplification  (3.14)  was  possible  because  the  damping  matrix  is,  after 
transformation  into  modal  coordinates,  equal  to  a  scalar  matrix. 

Now,  following  Hayashi  [7,  page  93],  we  will  express  the  solution  of  (3.14)  in 
the  form: 

|6f|  =  e^({bi)cos(®t)  +  {aj}sin(®t))  (3.15) 

which  should  allow  us  to  determine,  in  a  first  approximation,  the  first  order  simple 
unstable  region. 

Inserting  (3.15)  into  (3.14)  and  appl3dng  the  HBM,  we  find 

(X^[l]  +  X[M,]  +  [M,]){y  =  {°}  (3.16) 


0  2cd[I] 
[-2cd[I]  0  J 


[Mo]  = 


where 
[M,]  = 


(3.17) 


^  "1  p'" 


[BriDjB] 

[Bf[D„lB]-L^+ril]1[l]^[a>o/] 


(3.18) 


To  determine  the  characteristic  exponents,  X,  we  transform  this  system  into  [16] 


0  [I]  Ifxl 

-[M„]  -[M,]JlrJ 


(3.19) 


where  {X}  is  a  vector  formed  by  {bi},  {aj.  The  values  of  X  are  the  eigenvalues  of 
the  double  size  matrix  in  the  previous  equation.  Bearing  in  mind  that  it  is  the  stability 
of  the  variable  {5^}  in  which  we  are  interested  we  substitute  equation  (3.15)  into 
equation  (3.13)  to  obtain 

{61}  =  {{b, }  cos(0t)  +  {a,  }sin((Dt)} .  (3.20) 


If  the  real  part  of  X^  - 


ii 

2  © 


is  positive  for  any  then  the  solution  is  unstable, 


otherwise  it  is  stable. 


1398 


For  undamped  systems,  it  was  demonstrated  in  [8],  that  important  conclusions 
about  the  stability  of  the  solutions  can  be  deducted  from  the  determinant  of  the 
Jacobian  of  {F}.  We  are  going  to  extend  the  demonstration  to  systems  with  mass 
proportional  damping. 

Applying  the  derivation  rule  for  composite  functions,  we  obtain  the  derivatives 
of  {Fj}  and  {fJ  with  respect  to  {wc}  and  {wj  as  follows; 


Matrices  [I]  and  [Mo]  are  symmetric  and  matrix  [Mi]  is  skew-symmetric.  This 
means  that  the  eigenvalues  of  equation  (3.16)  are  either  purely  imaginary  or  purely  real 
[8].  If  X  is  imaginary  the  solution  is  always  stable;  if  X  is  real  the  stability  limit  is 
defined  by 

=  (3.25) 


Inserting  (3.25)  in  (3.16)  we  arrive  at 

■[Bf[D„][B]-co^[l]  +  [o.,/]  [Bf[D.][B]  +  P  l|b,l  Jol 

[Bf[D„][B]-p  [BnD,,][B]-<B^[l]  +  [co,/]JlaJ  loj 

The  matrix  in  the  previous  equation  is  [B]^[D][B],  where  [D]  is  the  Jacobian 
of  {F’}  with  respect  to  the  vector  of  coefficients  of  the  cosine  and  sine  terms,  given  by 


[D]  =  d{F}/d 


A  non-trivial  solution  of  (3.26)  exists  if 

det([Bf  [D][B])=0  o  |BnDl=0o|Dl=0.  (3.28) 

The  last  equivalence  is  true,  because  [B]  is  a  non-singular  matrix.  Thus,  we  proved  that 
in  the  stability  limit,  the  determinant  of  the  Jacobian  of  (F),  |D|  ,  is  zero. 


|D|  is  a  polynomial  in  the  coefficients  {wj  and  (wj  and  in  o;  therefore,  it  is  a 

continuous  function  in  those  coefficients.  All  the  experimental  and  numerical  analysis 
of  nonlinear  vibration  of  beams,  indicate  that  the  shape  of  vibration,  defined  in  our 
model  by  {wc}  and  {w^},  is  a  continuous  function  of  the  amplitude  and  the  frequency 
of  vibration.  Thus,  we  conclude  that  |d|  varies  in  a  continuous  way  through  the  FRF 
(fi-equency  response  function)  curve.  Hence,  if  there  is  a  change  in  its  sign  between 
two  consecutive  points  of  the  FRF  curve,  then  |D|  =0  for  a  particular  point  between 

these  two.  In  that  particular  point,  the  stability  limit  might  have  been  crossed. 

So,  a  complete  study  of  the  first  order  solution’s  stability  may  be  carried  out  by 
determining  only  the  characteristic  exponents  of  the  first  solution  and  when  JD]  changes 
sign  or  when  |D|  is  approximately  zero.  As  jD|  is  needed  in  the  continuation  method  and, 
when  the  Newton  method  is  applied,  can  be  easily  calculated  from  [D],  this  results  in 
substantial  time  savings. 


4 -APPLICATIONS 

A  clamped-clamped  beam  made  in  an  aluminium  alloy  with  the  reference  7075- 
T6  was  analysed.  Its  material  [9]  and  geometric  properties  are: 

E  =  7. 172*10^°  N/m2,  p  =  2800  kg/m^,  h=0.002  m,  b=0.02  m,  L  =  0.405  m  3. 

For  aluminium,  a  typical  value  of  the  loss  factor  (which  is  multiplied  by  the 
stiffness  matrix)  is  a=0.01  (C=0.5%),  but  the  measured  value  in  reference  [18]  was 
approximately  equal  to  0.038  (^sl.9%)^.  In  order  to  have  the  same  damping 
coefficient  for  the  first  mode  of  vibration,  the  value  of  the  damping  factor  is: 

P  =  (»o,^xa.  (4.1) 

The  beam  was  modelled  using  the  HFEM,  as  described  in  section  2.  To  solve 
the  system  of  equations  (2. 11),  Newton’s  method  was  used  in  the  nonresonant  area.  In  the 
vicinity  of  resonance  frequencies  it  is  difficult  to  obtain  convergence  by  the  Newton  method 


3  Except  in  the  comparison  with  experimental  results,  where  L-0.406  (value  of  Wolfe’s  clamped- 
clamped  beam  length). 

Wolfe  did  not  think  that  the  measured  damping  ratio  was  only  due  to  material  damping.  He  also 
attributed  the  obtained  value  to  damping  in  the  joints  and  to  the  coil  magnet  arrangement  used  to 
excite  the  beam. 


1400 


and  a  continuation  method  was  applied  [8,  14].  The  derivation  of  the  Jacobian  matrix 
present  in  both  methods  was  performed  symbolically  [15]. 

Because  the  excitation  force  will  be  applied  in  the  middle  of  the  beam  and  both 
the  beam  and  the  boundary  conditions  have  symmetric  properties,  only  symmetric  out 
of-plane  shape  functions  and  antisymmetric  in-plane  shape  functions  will  be  used^. 

4.1  -  Study  of  convergence  with  number  of  shape  functions 

With  four  out-of-plane  (po)  and  four  in-plane  shape  functions,  convergence  of 
the  value  of  the  first  linear  natural  frequency  is  achieved  (Table  1).  This  number  of 
shape  functions  will  be  the  starting  value  for  our  nonlinear  analysis.  The  number  of 
degrees  of  freedom  of  the  present  damped  model  is  equal  to  2xpo. 


Table  1  -  Natural  linear  frequencies  of  the  cc  beam  (rad/s).  Mode  1. 


Exact 

Po=2,  pi=2 

Po=3,  pi=3 

Po=4,  p=4i 

Po=5,  Pi=5 

396.6 

396.613  239 

396.  605  011 

396.  605  008 

396.  605  008 

In  Figures  1,  2  and  3  we  can  see  the  FRFs  in  the  vicinity  of  the  first,  third  and 
fifth  mode,  obtained  when  a  force  P  of  0.03  N  was  applied.  Near  the  first  mode  there  is 
no  increase  in  accuracy  by  using  more  than  four  out-of-plane  and  four  in-plane  shape 
functions  (po=4,  pi=4).  However,  for  the  third  mode,  as  the  amplitude  of  vibration 
grows,  the  results  obtained  with  po=4  and  pi=4  depart  from  the  ones  obtained  with 
more  shape  functions.  The  FRF  curve  constructed  with  po=5  and  pi=5  is  quite  similar 
to  the  coincident  FRFs  obtained  with  po=6,  pi=6  and  with  po=7,  pi=7.  In  the 
neighbourhood  of  the  fifth  mode,  convergence  seems  to  be  achieved  with  po=8  and 

pi=8. 


4.2  -  Influence  of  in-plane  displacements 

In  Figure  4  the  FRFs  obtained  considering  and  neglecting  the  in-plane 
displacements  are  compared.  As  in  references  [1]  and  [10],  we  found  that  the  in-plane 
displacements  ‘reduce’  the  non-linearity,  in  the  sense  that  the  non-linearity  caused  by 
them  is  of  the  soft  spring  type  and  counterbalances  the  hard  spring  type  non-lineanty 

5  To  check  if  the  nonlinearity  introduced  any  coupling  and  consequent  antisymmetric  terms  in  the 
response,  a  model  including  symmetric  and  antisymmetric,  in-  and  out-of-plane  shape  functions  was 
considered.  It  was  confirmed  that,  with  these  boundary  conditions  and  with  the  one  harmonic 
representation  of  the  solution’s  time  dependency,  there  is  no  such  coupling. 


1401 


caused  by  the  transverse  displacements.  This  ‘reduction’  of  nonlinearity  is  due,  as  the 
formulation  of  the  nonlinear  stiffness  matrix  -  eq.  (2.9)  -  shows,  to  the  effects  of  in¬ 
plane  deformation  on  the  stiffness  of  the  structure. 

4.3  -  Study  of  stability 

In  Figures  5  and  6  we  can  see  the  stability  studies  carried  out  in  the 
neighbourhood  of  the  first  and  third  resonance  frequencies,  using  po=6,  pi=6  shape 
functions  and  with  an  excitation  force  of  amplitude  P  =  0.03  N.  In  all  cases,  |D| 
changed  sign  when  the  stability  condition  of  the  solutions  changed. 

4.4  -  Comparison  with  experimental  results 

In  Figures  7  and  8  we  can  see  the  comparison  between  the  FRF  obtained  with 
the  HFEM,  using  po=6,  pj=6  shape  functions,  and  the  experimental  results  [18]  when  a 
force  P  =  0.134  N  is  applied  in  the  centre  of  the  beam.  Two  values  were  used  for  the 
damping  factor:  P=0.01g3q,  and  P=0.038g)o,  . 

The  HFEM  provides  a  FRF  with  a  slope  similar  to  the  experimental  one  around 
the  resonance  frequency.  This  indicates  that  the  nonlinear  stiffness  is  well  represented 
by  the  model. 

The  turning  point  corresponding  to  the  largest  amplitude  of  vibration,  where  the 
jump  phenomena  occurs,  obtained  with  the  HFEM,  point  B,  does  not  match  the 
experimental  one,  point  A.  With  the  typical  value  used  for  the  loss  factor  in  aluminium 
alloys,  a=0.01  (p=0.01oi)oj),  the  maximum  amplitude  of  vibration  was  more  than  double 
the  one  measured.  However,  the  HFEM  solutions  represented  in  Figure  7  after  point  A  are 
very  close.  Thus,  in  a  real  system,  a  small  perturbation  would  easily  make  the  shape  of 
vibration  change  into  an  unstable  one  and  a  change,  or  jump,  to  a  stable  shape  of  vibration 
at  a  lower  amplitude  could  be  observed  before  the  largest  computed  amplitude  of  vibration 
was  achieved.  With  the  measured  loss  factor,  a=0.038,  the  largest  amplitude  of  vibration 
obtained  with  the  HFEM,  was  around  a  half  of  the  measured  maximum  amplitude. 


1402 


5  -  CONCLUSIONS 


The  HFEM  dynamic  model  of  a  beam  vibrating  with  large  amplitudes  was 
constructed  with  small  time  expense.  This  is  due  to  the  small  number  of  degrees  of  freedom  with 
which  convergence  is  achieved,  to  the  easy  way  in  which  the  number  of  d.o.f  are  reduced, 
benefiting  from  the  symmetry  properties  of  the  problem,  and  to  the  embedding  properties^  of 
the  HFEM. 

For  the  amplitudes  of  vibration  displacement  studied,  with  relatively  few  d.o.f 
the  FRF  curves  were  accurately  determined  until  the  5th  order  mode,  inclusive.  If 
modes  of  order  higher  than  5th  are  to  be  studied,  then  the  inclusion  of  more  elements 
instead  of  more  shape  functions  should  be  considered,  as  shape  functions  of  excessive 
high  order  turn  the  construction  of  the  matrices  quite  time  consuming.  The  comparison 
with  experimental  results  showed  a  very  good  prediction  of  the  slope  of  the  FRF  by  the 
HFEM.  The  largest  amplitude  of  vibration  and  the  correspondent  turning  and  jump 
point,  are  greatly  influenced  by  the  amount  of  damping  used. 

Using  the  flexibility  of  choosing  the  shape  functions  in  the  HFEM  model  it  was 
shown  that  the  in-plane  displacements  cause  a  softening-type  nonlinearity. 

With  the  continuation  method  the  multi-valued  regions  of  the  FRF  curves  were 
completely  and  automatically  described. 

To  determine  the  characteristic  exponents  that  establish  the  stability  of  the 
solution,  we  solved  an  eigenvalue  problem.  Due  to  the  reduced  number  of  degrees  of 
freedom  of  the  HFEM  model  this  was  quickly  solved.  More  important,  it  was  proven 
that  in  the  stability  limit  the  determinant  |D|  is  zero.  Thus,  we  only  have  to  determine 
the  characteristic  exponents  of  the  first  solution  and  when  there  is  an  indication  that 
|Dj  =0  for  a  particular  point,  to  check  if  the  stability  of  the  solution  changed.  This 

results  in  significant  time  savings. 

With  symbolic  computation,  the  matrices  involved  in  the  HFEM  model  and  the 
Jacobian  matrix  necessary  in  the  continuation  and  Newton  methods,  were  easily  and 
exactly  derived,  thus  reducing  the  numerical  errors.  Symbolic  computation  was  also 
helpful  in  the  application  of  the  HBM. 


^  Here  we  include  the  HFEM’s  advantages  in  the  derivation  of  the  nonlinear  stiffness  matrix. 


1403 


REFERENCES 


[  1  ]  -  Atluri,  S.,  Nonlinear  vibrations  of  a  hinged  beam  including  nonlinear  effects.  Trans,  of 
the  ASMS  J.  of  Apl.  Mech.,  1973,  40,  121-126. 

[  2  ]  -  Bermet,  J.  A.  and  Eisley,  J.  G.,  A  multiple-degree-of  freedom  approach  to  nonlinear 
beam  vibrations.  J.  of  the  Am.  Inst,  of  Aeronaut,  and  Astronaut.,  1970, 8,  734-739. 

[  3  ]  -  Cheung,  Y.  K.  and  Lau,  S.  L.,  Incremental  time-space  finite  strip  method  for  non-linear 
structural  vibrations.  Earthquake  Engng.  and  Struct.  Dynamics,  1982, 10, 239-253. 

[  4  ]  -  Han,  W,  The  Analysis  of  isotropic  and  laminated  rectangular  plates  including 
geometrical  non-linearity  using  the  p-version  finite  element  method^  Ph.D.  Thesis,  University 
of  Southampton,  Southampton,  1993. 

[  5  ]  -  Hamdan,  M.N.  and  Burton,  T.D.,  On  the  steady  state  response  and  stability  of  non¬ 
linear  oscillators  using  harmonic  balance.  J.  of  Sound  and  Vibr.,  1993, 166,  255-266. 

[  6  ]-  Stokey,  W.  F.,  Shock  and  Vibration  Handbook,  Third  edition,  ed.  C.  M.  Harris, 
McGraw-Hill,  New  York,  1988,  p.  7-14. 

[  7  ]  -  Hayashi,  C,  Nonlinear  Oscillations  in  Physical  Systems,  McGraw-Hill,  New  York, 
1964. 

[  8]  -  Lewandowski,  R.,  Non-linear,  steady-state  analysis  of  multispan  beams  by  the  finite 
element  method.  Computers  and  Struct.,  1991, 39,  83-93. 

[  9  ]  -  ASM  Committee  on  Properties  of  Aluminium  Alloys,  Properties  and  Selection  of 
Metals,  Metals  Handbook,  Vol.  1,  8th  edition,  ed.  T.  Lyman,  Ohio,  1961,  p.  948. 

[  10  ]  -  Mei,  C  and  Decha-Umphai,  A  finite  element  method  for  non-linear  forced  vibrations  of 
beams.  J.  of  Sound  and  Vibr.,  1985, 102,  369-380. 

[  11  ]  -  Meirovitch,  L.,  Elements  of  Vibration  Analysis,  McGraw-Hill,  Singapore,  1986. 

[  12  ]  -  Mentel,  T.  J.,  Vibrational  energy  dissipation  at  structural  support  junctions.  In 
Colloquium  on  Struct.  Damping,  ed.  E.  J.  Ruzicka,  1959,  pp.  89-116. 

[  13  ]  -  Petyt,  M,  Introduction  to  Finite  Element  Vibration  Analysis,  Cambridge  University 
Press,  Cambridge,  1990. 

[  14  ]  -  Ribeiro,  P.  and  Petyt,  M.,  Study  of  nonlinear  free  vibration  of  beams  by  the 
hierarchical  finite  element  method.  ISVR  Techmcal  Memorandum  No. 773,  University  of 
Southampton,  Southampton,  November  1995. 

[  15  ]  -  Redfem,  Darren,  The  Maple  Handbook,  Springer-Verlag,  New  York,  1994. 

[  16  ]  -  Takahashi,  K.,  A  method  of  stability  analysis  for  non-linear  vibration  of  beams.  J.  of 
Sound  and  Vibr.,  1979,  67,  43-54. 

[  17  ]  -  Wang,  S.  S.  and  Huseyn,  K.,  Bifurcations  and  stability  properties  of  nonlinear  systems 
with  symbolic  software.  Math.  Comput.  Modelling,  1993, 18,  21-38. 

[  18  ]  -  Wolfe,  Howard,  An  experimental  investigation  of  nonlinear  behaviour  of  beams  and 
plates  excited  to  high  levels  of  dynamic  response,  Ph.D.  Thesis,  University  of  Southampton, 
Southampton,  1995. 


1404 


FIGURES 


mnxiwi  1 

h 

0.5- 

€f^ 

().4- 

0.3- 

.  g 

0.2- 

« 

O.I- 

.  • 

O' 

340  360  380  4^0  420 

0)  (rad/s) 

1  -  FRF  in  the  vicinity  of  the  first  mode  of  vibration.  x=0.5 
0  po=4,  pi=4;  □  po=5,  pi=5;  0  po=6,  pi=6;  +  po=7,  pi=7. 


1405 


innxlw 

h 

1 

0.5- 

9°’ 

0.4- 

t  ,  . 

0.3- 

0.2 

% 

0.1 

. 

. 

-  *  *  ‘  . 

0^ 

340 

360 

380  400  420  440 

CO  (rad/s) 

Figure  4  -  FRF  with  in-plane  displacements,  po=6,  Pi=6  (o),  and  without 
in-plane  displacements,  po=6,  Pi=0  (+).  x=05xL. 


Figure  5  -  Stability  study.  First  mode.  x=0.5xL.  □  stable  solution; 
+  unstable  solution;  po=6,  pi=6. 


1406 


mnxiwi 

h 

B 

2.2- 

2- 

1.8- 

1.6- 

1.4- 

1,2- 

I- 

A 

0.8 

0.6- 

0.4' 

0.8  1  .2  ,  1.4  1.6  1.8 

CO/OD„ 

0 

Figure  7  ~  Comparison  with  experimental  results,  o  HFEM  stable, 

□  HFEM  unstable,  Po=6  and  pi=6,  |3=0,01cOo ,2;  +  experimental.  x=05xL. 


mnxiw 

h 

1- 

A 

0.8- 

* 

0.6 

B 

0.4 

O.2J 

. 

• 

* 

0.8  0.9  1  1.1 

M/Wq 

1.2 

1.3 

Figure  8  -  Comparison  with  experimental  results,  o  HFEM  stable, 

□  HFEM  unstable,  po=6  and  pi=6,  (5=0.038cOo,2;  +  experimental.  x=05xL. 


1407 


1408 


GEOMETRICALLY  NONLINEAR  DYNAMIC  ANALYSIS 
OF  3-D  BEAM 


Kuo  Mo  Hsiao  and  Wen  Yi  Lin 
Department  of  Mechanical  Engineering, 
National  Chiao  Tung  University, 
Hsinchu,  Taiwan,  Republic  of  China 


ABSTRACT 

A  co-rotational  finite  element  formulation  for  the  geometrically 
nonlinear  dynamic  analysis  of  spatial  beam  with  large  rotations  but 
small  strain  is  presented.  The  deformation  nodal  forces  and  inertia 
nodal  forces  are  derived  by  using  the  d'Alembert  principle  and  the 
virtual  work  principle.  The  gyroscopic  effect  is  considered  here. 

The  beam  element  developed  here  has  two  nodes  with  sbc  degrees 
of  freedom  per  node.  Some  angular  velocity  coupling  terms,  which 
are  so  called  gyroscopic  forces,  are  obtained  in  inertia  nodal  force. 

An  incremental-iterative  method  based  on  the  Newmark  direct 
integration  method  and  the  Newton-Raphson  method  is  employed 
here  for  the  solution  of  the  nonlinear  dynamic  equilibrium 
equations.  Numerical  examples  are  presented  to  demonstrate  the 
acctuacy  and  efficiency  of  the  proposed  method. 

INTRODUCTION 

In  recent  years,  the  nonlinear  dynamic  behavior  of  beam 
structures,  e.g.,  framed  structures,  flexible  mechanisms,  and  robot 
aims,  has  been  the  subject  of  considerable  research.  In  [1],  Hsiao  ^d 
Jang  presented  a  co-rotational  formulation  and  numerical 
procedure  for  the  dynamic  analysis  of  planar  beam  structures.  This 
formulation  and  numerical  procedure  were  proven  to  be  very 
effective  by  numerical  examples  studied  in  [1].  However,  it  is  only 
limited  for  planai*  beam  structures.  A  general  formulation  for  three 
dimensional  beam  element  is  not  a  simple  extension  of  a  two 
dimensional  formulation,  because  large  rotations  in  three 
dimensional  analysis  are  not  vector  quantities;  that  is,  they  do  not 
comply  with  the  rules  of  vector  operations.  In  [2]  a  motion  process 
of  the  three  dimensional  beam  element  is  proposed  for  the  large 
displacement  and  rotation  analysis  of  spatial  frames.  In  [3]  a  co- 
rotational  formulation  for  three-dimensional  beam  element  is 
proposed.  However,  it  is  only  limited  for  nonlineai'  static  analysis. 

The  objective  of  this  study  is  to  present  a  practical  formulation  for 
the  dynamic  analysis  of  three  dimensional  Euler  beam.  The 
kinematics  of  the  beam  element  proposed  in  [3]  is  adopted  here. 


1409 


The  element  deformations  are  determined  by  the  rotation  of 
element  cross  section  coordinates,  which  are  rigidly  tied  to  element 
CTOSS  section,  relative  to  the  element  coordinate  system  [2,  3].  The 
three  rotation  parameters  proposed  in  [3]  are  used  to  determine  the 
orientation  of  the  element  cross  section  coordinates.  In  order  to 
capture  the  gyroscopic  effect,  the  relation  between  the  time 
derivatives  of  the  rotation  parameters  and  the  angular  velocity  and 
the  angular  acceleration  is  derived  here.  The  beam  element 
developed  here  has  two  nodes  with  six  degrees  of  freedom  per  node. 
The  element  nodal  forces  are  conventional  forces  and  moment. 
The  deformation  nodal  forces  and  inertia  nodal  forces  are  derived 
by  using  the  d'Alembert  principle  and  the  virtual  work  principle  in 
the  current  element  coordinates.  An  incremental-iterative  method 
based  on  the  Newmark  direct  integration  method  and  the  Newton- 
Raphson  method  is  employed  here  for  the  solution  of  the 
nonlinear  dynamic  equilibrium  equations.  Numerical  examples 
are  presented  to  demonstrate  effectiveness  of  the  proposed  method. 

FINITE  ELEMENT  FORMULATION 
Basic  assumptions 

The  following  assumptions  are  made  in  the  derivation  of  the 
nonlinear  behavior:  (1)  the  beam  is  prismatic  and  slender,  and  the 
Euler-Bernoulli  hypothesis  is  valid;  (2)  the  centroid  and  the  shear 
center  of  the  ctoss  section  coincide;  (3)  the  unit  extension  and  twist 
rate  of  the  centroid  axis  of  the  beam  element  are  uniform;  (4)  the 
cross  section  of  the  beam  element  does  not  deform  in  its  own  plane, 
and  strains  within  this  cross  section  can  be  neglected;  (5)  the  out-of¬ 
plane  warping  of  the  cross  section  is  the  product  of  the  twist  rate  of 
the  beam  element  and  the  Saint  Venant  warping  function  for  a 
prismatic  beam  of  the  same  cross  section;  (6)  the  deformations  of  the 
beam  element  are  small. 

Coordinate  systems 

In  this  paper,  a  co-rotational  total  Lagrangian  formulation  is 
adopted.  In  order  to  describe  the  system,  following  [3],  we  define 
three  sets  of  coordinate  systems  (see  Fig.  1): 

(1)  A  fixed  global  set  of  coordinates,  Xj(z  =  1,2,3);  the  nodal 
coordinates,  displacements,  and  rotations,  and  the  stiffness  matrix 
of  the  system  are  defined  in  this  coordinates. 

(2)  Element  cross  section  coordinates,  rf  (f  =  1,2,3);  a  set  of  element 
cross  section  coordiaates  is  associated  with  each  cross  section  of  the 
beam  element.  The  origin  of  this  coordinate  system  is  rigidly  tied  to 


1410 


the  shear  center  of  the  cross  section.  The  xf  axes  are  chosen  to 
coincide  with  the  normal  of  the  corresponding  cross  section  and  the 

X2  and  X3  axes  are  chosen  to  be  the  principal  directions  of  the  cross 
section. 

(3)  Element  coordinates,  x,  (z  =  1,2,3);  a  set  of  element  coordinates 
associated  with  each  element.  The  origin  of  this  coordinate  system 
is  located  at  node  1;  the  xj  axis  is  chosen  to  pass  through  two  end 
nodes  of  the  element,  and  the  X2  and  X2  axes  are  determined  from 
the  orientation  of  the  element  cross  section  coordinates  at  two  end 
nodes  using  the  way  given  in  [2].  The  deformations  and  stiffness 
matrices  of  the  elements  are  defined  in  terms  of  this  coordinates. 
In  this  paper  the  element  deformations  are  determined  by  the 
rotation  of  element  cross  section  coordinates  relative  to  this 
coordinate  system. 

Rotation  vector  and  rotation  parameters 
For  convenience  of  the  later  discussion,  the  term  'rotation  vector' 
is  used  to  represent  a  finite  rotation.  Figure  2  shows  that  a  vector 
b  which  as  a  result  of  the  application  of  a  rotation  vector  (pa.  is 
transported  to  the  new  position  h'.  The  relation  between  b  and  b' 
may  be  expressed  as  [4] 

b'  =  cos  0b  +  (1  -  cos  0)(a  •  b)  +  sin  0(a  x  b),  (1) 

where  (p  is  the  angle  of  counterclockwise  rotation,  and  a  is  the  unit 
vector  along  tiie  axis  of  rotation. 

In  this  paper,  the  s)mbol  {  }  denotes  column  matrix.  Let  e,-  and  ef 
(i  - 1, 2, 3)  denote  the  unit  vectors  associated  with  the  x,  and  xf  axes, 

respectively.  Here,  the  traid  ef  in  the  deformed  state  is  assumed  to 
be  achieved  by  the  application  of  the  following  two  rotation  vectors 
to  the  traid  Cj  : 

e„  =  0„n,  =  0it, 

where 

n = {0, 82!  (el  +  03/(0!  +  0|)V2} 

=  {0,^2,713}, 
t  =  {cos0„,  62,63}. 

cos0„=(i-0!-e!)Vl 

^  dw{s)  .  dv(s) 


(2,3) 


(4) 


1411 


in  which  n  is  the  unit  vector  perpendicular  to  the  vectors  ei  and 
ef,  and  t  is  the  tangent  unit  vector  of  the  deformed  centroid  axis. 

Note  that  ef  coincides  with  t.  is  the  inverse  of  cosd„.  v(s)  and 
w(s)  are  the  lateral  deflections  of  the  centroid  axis  of  the  beam 
element  in  the  X2  and  directions,  respectively,  and  s  is  the  arc 
length  of  the  deformed  centroid  axis. 

The  rotation  vectors  e„  and  0^  are  determined  by  (f  =  l,2,3). 
Thus,  di  are  called  rotation  parameters  in  this  study. 

Using  Eqs.  (l)-(4),  the  relation  between  the  vectors  and  ef 
(i  =  1,2,3)  in  the  element  coordinate  system  may  be  obtained  as 

ef  =  [t,  Ri,  R2]  =  Re,-, 

Rl  =  cos^iri  +  sin0ir2, 

R2  =  -sin^iri  +  cos0ir2, 

ri  =  {-03,  cos  0„  +  (1  -  cos  0„ ,  (1  -  cos  d„)n2n^h 
r2  ={02^(1-cos0„)w2W3,cos0„  +(l-cos0„)n3},  (5) 


where  R  is  the  so-called  rotation  matrix. 

Let  0  =  {01,  02,  03>  be  the  vector  of  rotation  parameters,  36  be  the 

variation  of  0.  The  traid  ef  corresponding  to  0  may  be  rotated  by  a 
rotation  vector  =  {3ipi,  3(l>2,  <5^3}  to  reach  their  new  positions 
corresponding  to  0  +  50  [3].  When  02  and  03  are  much  smaller 
than  unity,  the  relationship  between  50  and  5<^  may  be 
approximated  by 


r  1 


50  = 


-03 


^2 


03/2  -02/2' 
1  0 

0  1 


(6) 


If  both  sides  of  Eq.  (6)  is  divided  by  St,  the  first  time  derivative  of  0 
may  be  expressed  by 


r  1 


0  = 


-03 


02 


03/2  -  02/2- 

1  0 

0  1 


(j)  =  T^^ij), 


(7) 


where  the  symbol  (  )  denotes  differentiation  with  respect  to  time  t . 


1412 


=  1,2,3)  denote  the  angular  velocities  about  the  axes. 

From  Eq.  (7),  the  second  time  derivative  of  6  may  be  expressed  by 
may  be  expressed  as 

9  =  t-^(j)  +  T"^^,  (8) 


where  =  1,2,3)  denote  the  angular  accelerations  about  the  Xi 
axes. 

Nodal  parameters  and  forces 

The  global  nodal  parameters  for  the  system  of  equations 
corresponding  to  the  element  local  nodes  j  (j  -  1,  2)  are  ll^j,  the  Xj 
(i  =  1,2,3)  components  of  the  translation  vectors  at  nodes  j,  and 
the  Xi  (f  =  1,2,3)  components  of  the  rotation  vectors  at 
nodes  j.  Here,  the  values  of  are  reset  to  zero  at  current 
configuration.  Thus,  <50^,  the  variations  of  ^ij,  represent 
infinitesimal  rotations  about  the  Xi  axes  [3],  <i>ij  and  Oy  represent 
angular  velocities  and  angular  accelerations  about  the  Xj  axes, 
respectively.  Tlie  generalized  nodal  forces  corresponding  to  dOy  are 
the  conventional  moments  about  the  Xf  axes.  The  generalized 
nodal  forces  corresponding  to  dllij,  the  variation  of  Uy-,  are  the 
forces  in  the  X,  directions. 

The  element  employed  here  has  six  degrees  of  freedom  per  node. 
Two  sets  of  element  nodal  parameters  termed  'explicit  nodal 
parameters'  and  'implicit  nodal  parameters'  are  employed.  The 
explicit  nodal  parameters  of  the  element  are  used  for  the  assembly 
of  the  system  equations  from  the  element  equations.  Thus,  they 
should  be  consistent  with  the  global  nodal  parameters,  and  are 
chosen  to  be  Uij,  the  x,  (i  =  l,2,3)  components,  of  the  translation 

vectors  uj  at  nodes  j  (j  =  1,  2)  and  0y,  the  Xi  (i  =  1,2,3)  components 
of  the  rotation  vectors  (j)j  at  nodes  j.  Similarly,  the  generalized 
nodal  forces  corresponding  to  Wy  and  d^ij  are  /y  and  my,  the  forces 
in  the  Xi  directions  and  the  conventional  moments  about  the  Xj 
axes,  respectively. 

The  implicit  nodal  parameters  of  the  element  are  used  to 
determine  the  deformation  of  the  beam  element.  They  are  chosen 
to  be  My,  the  Xj  (i  =  l,2,3)  components  of  the  translation  vectors  u^ 
at  nodes  j  and  0y,  the  nodal  values  of  the  rotation  parameters  0- 


1413 


(i  =  1,2,3)  at  nodes  ;  (j  =  1,  2).  The  generalized  nodal  forces 
cori'esponding  to  duij  and  dOij  are  and  m|,  the  forces  in  the  Xi 
directions  and  the  generalized  moments,  respectively.  Note  that 
are  not  conventional  moments,  because  S6ij  are  not 

infinitesimal  rotations  about  the  Xi  axes. 

In  view  of  Eq.  (6),  the  relations  between  the  variation  of  the 
implicit  and  explicit  nodal  parameters  may  be  expressed  as 


dui' 

1 

0 

0 

0 

dui 

dQi 

0 

TI^ 

0 

0 

5<i>i 

'  dU2 

►  — 

0 

0 

I 

0 

5u2 

502 

0 

0 

0 

Ti\ 

^^2 

(9) 


where  daj  =  {duij,du2j,Su3j},  d6j={ddij,5d2j,862j},  and 
%  ={8<hj’^<p2j>^<p3jh  (j  =  %  2).  I  and  0  are  the  identity  and  zero 

matrices  of  order  3x3,  respectively.  (j  =  1,  2)  are  nodal  values 

of  T"l 

Let  f  =  {fi,mi,f2,m2K  f®  ={fi,mf,f2,m^},  where  ij  ={fij,f2jj3jh 
and  mj  (/’  =  h  2),  denote  the 

internal  nodal  force  vectors  corresponding  to  the  variation  of  the 
explicit  and  implicit  nodal  parameters,  dq  and,  <5q^,  respectively. 
Using  the  contragradient  law  [5]  and  Eq.  (7),  the  relation  between  f 

and,  f  ^  may  be  given  by 

f  =  (10) 


Kinematics  of  beam  element 

The  deformations  of  the  beam  element  are  described  in  the  current 
element  coordinate  system.  From  the  kinematic  assumptions  made 
in  this  paper,  the  deformations  of  the  beam  element  may  be 
determined  by  the  displacements  of  the  centroid  axis  of  the  beam 
element,  orientation  of  the  cross  section  (element  cross  section 
coordinates),  and  the  out-of-plane  warping  of  the  cross  section  [3]. 
Let  Q  (Fig.  1)  be  an  arbitrary  point  in  the  beam  element,  and  P  be  the 
point  corresponding  to  Q  on  the  centroid  axis.  The  position  vector 
of  point  Q  undeformed  and  deformed  configurations  may  be 

expressed  as 


1414 


and 


r^xei  +  ije2  +  ze^, 


(11) 


r  =  +  ^(s)e2  +  ^(s)e3  +  yel  +  ze|  +  Si^s^Qv  (1^) 

where  Xf,{s),  v{s)  and  w{s)  are  the  xy  X2  and  X't,  coordinates  of  point 
P,  respectively,  s  is  the  arc  length  of  the  deformed  centroid  axis 
measured  from  node  1  to  point  P.  ^c(®)  expressed  by 

Xc{s)  =  Mil  + 

where  un  is  the  displacement  of  node  1  in  the  xi  direction,  and 
cos0„  is  defined  in  Eq.  (4). 

Here,  z;(s)  and  w{s)  in  Eq.  (12)  are  assumed  to  be  the  Hermitian 
polynomials  of  s  ,  and  0i(s)  in  Eq.  (12)  is  assumed  to  be  linear 
polynomials  of  s,  and  maybe  given  by 

z;(s)  =  {Ni,N2,N3,N4}^{W21/%1/W22/%2}-= 

w{s)  =  {N'i,-N2,N3-,N4}^{m31,021/^32^^22}  = 

0l(s)  =  {N5,N6}^{0n^%>  =  N^u^,  (14) 

Ni  =  7(1  -  ?)^(2  f  I),  Nj  =  |(1  - 1^)(1  - 1). 

4  o 

yvj  =  i(i  + 1)2(2 -  f),  Ni  =  |(-1  +  ?2)(1  + 1). 

4  o 

Af5  =  |(l-?).  W6=|(l  +  |),  (15) 

where  S  is  the  arc  length  of  the  centroid  axis  of  the  beam  element 
and  may  be  expressed  by 

S^ltj QOS (17) 

where  I  is  the  chord  length  of  the  centroid  axis  of  the  beam 
element,  and  cos  is  given  in  Eq.  (4). 

The  way  to  determine  the  current  element  cross  section 


1415 


coordinates  at  both  ends,  element  coordinates,  and  element  implicit 
nodal  parameters  corresponding  to  displacement  increments  is 
given  in  [2, 3]. 

If  ;c,  y,  and  z  in  Eq,  (11)  are  regarded  as  the  Lagrangian 
coordinates,  the  Green  strains  en,  Si2,  and  613  are  given  by 


£11  = 


1 

2 


\dxj  [dxj 


-1 


£12 


2[^xJ 


2[^xJ  \dzj' 


(18) 


Substituting  Eqs.  (4),  (5),  (12),  and  (13)  into  Eq.  (18),  sn,  ei2,  and  £13 
can  be  calculated. 

Element  nodal  force  vector 

The  element  nodal  force  vector  (Eq.  (10))  corresponding  to  the 
implicit  nodal  parameters  are  obtained  from  the  d'Alembert 
principle  and  the  virtual  work  principle.  For  convenience,  the 
implicit  nodal  parameters  are  divided  into  four  generalized  nodal 
displacement  vectors  u,-  (i  =  a,b,c,d),  where 

Urt  «{«!!,  «i2}/ 

and  u^,  Uc,  and  u^  are  defined  in  Eq.  (14).. 

The  generalized  force  vectors  corresponding  to  (5Uj,  the  variation 
of  u,-  (f  =  a,h,c,d)  are 


4  -  +  4  = 


where  f^  and  f*-  {i  =  a,b,c,d)  are  the  deformation  nodal  force  vector 
and  the  inertia  nodal  force  vector,  respectively. 

The  virtual  work  principle  requires  that 


1416 


du^Ja  +  <50^4  +  dn^ic  + 

= JL  {oiideii  +  2ai2dei2  +  2ai3(5ei3  +  p8r^r)dV , 

where  on  =  Esiy  O12  =  2Gei2  and  0-13  =  2Gei2,  where  E  is  tl^ 
Young's  modulus  and  G  is  shear  modulus,  p  is  the  density,  and  V 
is  the  volume  of  the  undeformed  beam. 

If  the  element  size  is  properly  chosen,  the  values  of  the  nodal 
parameters  (displacements  and  rotations)  of  the  element  defined  in 
the  current  element  coordinate  system,  which  are  the  total 
deformational  displacements  and  rotations,  may  always  be  much 
smaller  than  unity.  Thus  only  the  first  order  terms  of  nodal 
parameters  are  retained  in  deformation  nodal  forces.  However,  in 
order  to  include  the  effect  of  axial  force  on  the  lateral  forces,  a 
second  order  term  of  nodal  parameters  is  retained.  Because  the 
values  of  the  nodal  parameters  of  the  element  may  always  be  much 
smaller  than  unity,  it  is  reasonable  to  assume  that  the  coupling 
between  the  nodal  parameters  and  their  time  derivatives  are 
negligible.  Thus  only  zeroth  order  terms  of  nodal  parameters  are 
retained  in  inertia  nodal  forces. 

From  Eqs.  (4),  (5),  and  (12)-(21),  the  deformation  nodal  forces  and 
the  inertia  nodal  forces  may  be  expressed  as 


Li 

(22, 23) 

.  .  rd  Gj{di2-Qn)f  1 11 

f“  =  +  k^)Uc,  td-  ^  i 

(24,  25) 

fa  = 

(26) 

4  =  +  m^)Ub  -  2p4  , 

(27) 

fc  =  -  2ply  fN'cdidsds  , 

(28) 

f^  =  m^Urf  -  p{ly  - 

(29) 

1417 


where  A  is  title  cross  section  area,  L  is  the  initial  length  of  the  beam 
element,  k,-  and  k^-  {i  =  b,c)  are  bending  and  geometric  stiffness 
matrices  of  conventional  beam  element  [5,6],  and  J  is  the  torsional 
constant  p  is  the  density,  ly  and  7^  are  the  moment  of  inertia  of 

the  beam  cross  section  about  the  and  axes  respectively,  m^  is 
the  consistent  mass  matrix  of  bar  element  for  axial  translation,  nij-f 
and  m,>  {i^b,c)  are  the  consistent  mass  matrices  of  elementary 
beam  element  for  lateral  translation  and  rotation,  respectively,  and 
is  the  consistent  mass  matrix  of  bar  element  for  axial  rotation. 
These  mass  matrices  can  be  found  in  [5,  6].  The  underlined  terms  in 
Eqs.  (27)-(29)  are  inertia  forces  induced  by  the  gyroscopic  effect,  and 
are  called  gyroscopic  forces. 

Element  Matrices 

The  element  stiffness  matrices  and  mass  matrices  may  be  obtained 
by  differentiating  the  element  nodal  force  vectors  with  respect  to 
nodal  parameters,  and  time  derivatives  of  nodal  parameters. 
However,  element  matrices  are  used  only  to  obtain  predictors  and 
correctors  for  incremental  solutions  of  nonlinear  equations  in  this 
study.  Thus,  approximate  element  matrices  can  meet  these 
requirements.  The  stiffness  matrices  and  mass  matrices  of 
elementary  beam  element  given  in  [5,  6]  are  also  used  here. 

Equations  of  motion 

The  nonlinear  equations  of  motion  may  be  expressed  by 

F^  =  F^  +  F^-P  =  0  (30) 

where  F^  is  the  unbalanced  force  among  the  inertia  nodal  force  F^, 
deformation  nodal  force  F^,  and  the  external  nodal  force  P.  F^  and 

F^  are  assembled  from  the  element  nodal  force  vectors  in  Eq.  (10), 
which  are  calculated  using  Eqs.  (10)  and  (22)-(29)  first  in  the  current 
element  coordinate  system,  and  then  transformed  from  current 
element  coordinate  system  to  global  coordinate  system  before 
assemblage  using  standard  procedure. 

APPLICATIONS 

An  incremental  iterative  method  based  on  the  Newmark  direct 
integration  method  and  the  Newton-Raphson  method  is  employed 
here  for  the  solution  of  the  nonlinear  d5mamic  equilibrium 
equations. 


1418 


The  example  considered  is  a  right-angle  cantilever  beam  subjected 
to  an  out-of-plane  concentrated  load  as  shown  in  Fig.  3.  Four 
elements  are  used  for  discretization.  A  time  step  size  of  At  =  0.25^  is 
used.  The  cantilever  undergoes  a  finite  free  vibration  with 
combined  bending  and  torsion  after  the  removal  of  the  applied 
load;  the  time  histories  of  out-of-plane  displacements  of  the  elbow 
and  of  the  tip  are  given  in  Figs.  4  and  5.  It  is  seen  that  the  present 
results  are  in  excellent  agreement  with  those  given  in  [7]  and  [8]. 
However,  it  should  be  mentioned  that  the  beam  elements  used  in 

[7]  and  [8]  are  derived  using  fully  nonlinear  beam  theory  and  total 
Lagrangian  formulation.  Thus,  the  beam  elements  used  in  [7]  and 

[8]  are  much  more  complicated  than  that  proposed  here. 

CONCLUSIONS 

A  co-rotationai  finite  element  formulation  for  the  geometrically 
nonlinear  dynamic  analysis  of  spatial  beam  with  large  rotatio^  but 
small  strain  is  presented.  The  deformation  nodal  forces  and  inertia 
nodal  forces  are  derived  by  using  the  d'Alembert  principle  and  the 
virtual  work  principle.  The  gyroscopic  effect  are  considered  here. 

The  nodal  coordinates,  displacements,  rotations,  velocities, 
accelerations,  and  the  equation  of  motion  of  the  system  are  defined 
in  a  fixed  global  set  of  coordinates.  The  beam  element  developed 
here  has  two  nodes  with  six  degi'ees  of  fi^eedom  per  node.  The 
element  nodal  forces  are  conventional  forces  and  moments.  AU  of 
element  deformations  and  element  equations  are  defined  in  terms 
of  element  coordinates  which  are  constructed  at  the  current 
configuration  of  the  beam  element.  The  element  deformations  are 
determined  by  the  rotation  of  element  cross  section  coordinates, 
which  are  rigidly  tied  to  element  cross  section,  relative  to  the 
element  coordinate  system.  In  conjunction  with  the  co-rotational 
formulation,  the  higher  order  terms  of  nodal  parameters  in 
element  nodal  forces  are  consistently  neglected. 

An  incremental-iterative  method  based  on  the  Newmark  direct 
integration  method  and  the  Newton-Raphson  method  is  employed 
here  for  the  solution  of  the  nonlinear  dynamic  equilibrium 
equations.  Numerical  examples  are  presented  to  demonstrate  the 
accuracy  and  efficiency  of  the  proposed  method 

It  is  believed  that  the  co-rotational  formulation  for  3-D  beam 
element  presented  here  may  represent  a  valuable  engineering  tool 
for  the  d5mamic  analysis  of  spatial  beam  structures. 

ACKNOWLEDGMENT 

The  research  was  sponsored  by  the  National  Science  Council, 
Republic  of  China,  under  contract  NSC86-2212-E-009-006. 


1419 


REFERENCES 

1.  Hsiao,  KM.  and  Jang,  J.Y.,  .  Nonlinear  dynamic  analysis  of  elastic 
frames.  Computers  &  Structures,  1991,  33,  769-781. 

2.  Hsiao,  KM.  and  Tsay,  C.M.,  A  motion  process  for  large 
displacement  analysis  of  spatial  frames.  International  Journal  of 
Space  Structures,  1991,  6,  133-139. 

3.  Hsiao,  K.M.,  Corotational  total  Lagrangian  formulation  for 
three-dimensional  beam  element.  AIAA  Journal,  1992,  30,  797- 
804. 

4.  Goldstein,  H.,  Classical  Mechanics,  Addision-Wesley,  Reading, 
MA,1980. 

5.  Dawe,  D.J.,  Matrix  and  Finite  Element  Displacement  Analysis  of 
Structures,  Oxford  Univ.  Press,  New  York,  1984. 

6.  Hsiao,  I<.M.,  A  study  on  the  dynamic  response  of  spatial  beam 
structures.  NSC  82-0401-E009-081  Report,  National  Science 
Council,  Taiwan,  Republic  of  China,,  1993. 

7.  Simo,  J.C.  and  Vu-Quoc,  L.,  On  the  dynamics  in  space  of  rods 
undergoing  large  motions  -  A  geometrical  exact  approach. 
Computer  Methods  in  Applied  Mechanics  and  Engineering, 
1988,  66, 125-161. 

8.  lura,  M.  and  Atluri,  S.N.,  Dynamic  analysis  of  finitely  stretched 
and  rotated  three-dimensional  space-curved  beams.  Computers 
&  Structures,  1988,  29,  875-889. 


Fig.  1  Coordinate  systems. 


1420 


Fig.2  Rotational  vector 


Material  Properties: 
EA==10" 

EIy  =  EI^=GJ  =  10^ 

Ap  =  1 

ply=pl,=  10 


Time  History  of  Load: 


Fig.3  Right-angle  Cantilever  beam 


0  5  10  15  20  25  30 


Time 


Fig.  4  Displacements  in  the  X3  direction  at  point  B. 


Fig.  5  Displacements  in  the  X3  direction  at  point  A. 


1422 


Nonlinear  Response  of  Composite  Plates  to 
Harmonic  Excitation  Using  The  Finite  Element 
Time  Domain  Modal  Method 


Raymond  Y,  Y.  Lee,  Yucheng  Shi  and  Chuh  Mei 
Department  of  Aerospace  Engineering 
Old  Dominion  University,  Norfolk,  VA  23529-0247 

Abstract 

A  multimode  time  domain  formulation  based  on  the  finite  element  method  for  large 
amplitude  vibrations  of  thin  composite  plates  subjected  to  a  combined  harmonic  excitation 
and  thermal  load  is  presented.  By  using  the  modal  reduction  method,  the  system  equations  of 
motion  in  physical  coordinates  are  transformed  into  the  linear  modal  coordinates  and  the 
sizes  of  the  system  matrices  are  reduced  drastically.  The  reduced  system  modal  equations  can 
be  handled  easily  with  less  computational  efforts.  The  jBrequency-maximum  deflection 
relations  of  simple  harmonic,  superharmonic  and  subharmonic  responses  are  predicted  by 
choosing  suitable  initial  conditions.  The  procedure  for  the  selection  of  the  initial  conditions  is 
also  presented.  A  laminated  composite  plate  is  studied  in  great  detail.  External  loadings 
considered  are  harmonic  excitations  or  combined  harmonic  and  thermal  loads.  The  steady 
state  responses  of  the  linear  modal  coordinates  are  presented  in  details  at  several  frequencies. 
Their  phase  plots,  power  spectrums  and  time  domain  graphs  are  given  and  discussed . 


Introduction 

The  increase  use  of  advanced  composites  as  high  performance  structural 
components  necessitates  accurate  prediction  methods  which  reflect  their 
multilayered  anisotropic  behavior.  Thin  laminated  composite  plates  subjected  to 
severe  harmonic  lateral  loadings  are  likely  to  encounter  flexural  oscillations 
having  amplitudes  of  the  order  a  plate  thickness.  For  the  prediction  of  forced 
vibration  response,  the  multilayered  anisotropic  behavior,  the  complex 
boundary  conditions,  and  the  complex  loading  cases  such  as  the  present  of  the 
thermal  loads  make  the  problem  even  more  difficult.  Methods  of  analysis 
dealing  with  large  deflections  are  thus  becoming  increasingly  important. 

Whitney  and  Leissa  [1]  have  formulated  the  basic  governing  equations  for 
nonlinear  vibrations  of  heterogeneous  anisotropic  plates  in  the  sense  of  von 
Karman.  Based  on  those  equations,  a  number  of  classical  continuum 


1423 


approaches  exists  for  the  analysis  of  nonlinear  plate  behavior.  In  general,  the 
Galerkin’s  method  is  used  in  the  spatial  domain,  where  the  plate  deflection  is 
expressed  in  terms  of  one  or  more  linear  vibrational  mode  shapes;  and  various 
techniques  in  the  temporal  domain  such  as  the  direct  numerical  integration, 
harmonic  balance,  incremental  harmonic  balance,  perturbation,  and  multiple 
scales  methods,  to  cite  a  few,  are  employed.  Excellent  collections  of  classical 
continuum  solutions  and  reviews  on  geometrically  nonlinear  analysis  of 
laminated  composite  elastic  plates  are  given  by  Chia  [2,3]  and  Sathyamoothy 
[4].  The  internal  resonance  of  nonlinear  systems  has  been  thoroughly 
investigated  using  the  multiple  scales  by  Nayfeh  and  Mook  [5].  Most  recently, 
Wolfe  et  al.  [6]  have  reviewed  various  analytical  methods  and  have  obtained 
experiment  data  on  beams  and  plates  excited  sinusoidally  or  randomly.  Most  of 
the  classical  continuum  solutions  of  composite  plates  have  been  limited  to 
single-mode  approximation.  This  is  due  to  the  difficulties  in  obtaining  the 
general  Duffing-type  multiple-mode  equations  using  the  Galerkin’s  approach 
especially  for  arbitrarily  (unsymmetrically)  laminated  composite  plates  with 
complex  boundary  conditions. 

The  finite  element  method  has  proven  to  be  a  powerful  and  versatile 
approach  for  structural  problems  of  complex  geometries,  boundary  conditions, 
and  loadings.  Reddy  [7]  has  reviewed  the  application  of  finite  element  methods 
to  linear  and  nonlinear  anisotropic  composite  plate  problems.  In  this  paper,  the 
nonlinear  steady  state  periodic  responses  of  thin  rectangular  arbitrarily 
laminated  composite  plates  excited  sinusoidally  with  or  without  the  presence  of 
thermal  load  are  presented  using  the  finite  element  time  domain  modal  method. 
A  rectangular  composite  plate  is  studied  in  detail. 


Formulation 

The  finite  element  system  equations  of  motion  for  large  amplitude  vibrations 
of  a  thin  laminated  composite  plate  can  be  expressed  as 

[M]{w}+[C]{w}+([K]-[Knt]+[K1(W)]+[K2(W)]){W}  =  {P(t)}+{PT} 

(1) 

where  [M],  [C],  [K],  [Knt]  ,  {P(t)}  and  {Pt}  are  constant  matrices  and  vectors 
and  represent  the  system  mass,  damping,  linear  stiffness,  thermal  effort  and 


1424 


loads,  respectively;  and  [Kl]  and  [K2]  are  the  first  and  second  order  nonlinear 
stiffness  matrices  and  depend  linearly  and  quadratically  on  the  unknown 
structural  nodal  displacements  {W},  respectively.  The  derivation  of  the  element 
matrices  and  load  vectors  and  their  explicit  expressions  are  referred  to 
references  [8,9]. 

The  system  equations  of  motion  presented  in  eq.  (1)  are  not  suitable  for 
direct  numerical  integration  because:  a)  the  nonlinear  stiffness  matrices  [Kl] 
and  [K2]  are  functions  of  the  unknown  nodal  displacements,  and  (b)  the 
number  of  degrees  of  freedom  (DOF)  of  the  system  nodal  displacements  {W} 
is  usually  too  large.  Therefore,  eq.  (1)  has  to  be  transformed  into  the  modal  or 
generalized  coordinates  of  much  smaller  DOF.  Various  reduction  methods  for 
nonhnear  problems  have  been  summarized  in  an  excellent  review  article  by 
Noor  [10].  For  nonlinear  dynamic  problems,  the  base  vectors  need  updating 
using  the  modal  methods  presented  in  [10].  In  the  present  formulation,  the 
forced  general  Duffing-type  modal  equations  will  have  constant  nonlinear 
modal  stiffness  matrices,  therefore  updating  of  the  base  vectors  is  not  needed. 
This  is  accomplished  by  a  modal  transformation  and  truncation 


{W}  =  ^q,(t){(|)}">=[<I>]{q}  (2) 

r=l 

where  the  system  mode  shapes  are  the  solution  from  the  linear  eigen-problem 
cOf  [M]{(j)}^^^  =[K]{({)}^^\  The  nonlinear  stiffness  matrices  [Kl]  and  [K2]  in 
eq.  (1)  can  now  be  expressed  as  the  sum  of  the  products  of  modal  coordinates 
and  nonlinear  modal  stiffness  matrices  as 

[Kl]  =  ^qr[Kl«)W)]  (3) 

r=l 

and 

[K2]  =  ^  ^q^qs  [K2((j)<">(t)<^’ )] (4) 

r=l  s=l 


The  nonlinear  modal  stiffness  matrices  [Kl]^"^  and  [KZ]^"®^  are  assembled  from 
the  element  nonlinear  modal  stiffness  [kl]^''^  and  [k2]^^^^  as 


1425 


(5) 


([K1]">,[K2]'”>)  =  ^([kl]®  ,[k2]<“>) 

all  elements 
+  bdy.  conds. 

where  the  element  nonlinear  modal  stiffness  matrices  are  evaluated  with  known 
system  linear  mode  W.  Thus,  the  nonlinear  modal  stiffness  matrices 
and  are  constant  matrices.  Equation  (1)  is  thus  transformed  to  the 

forced  general  Duffing-type  modal  equations  as 

[M]{q}+[c]{q}+([K]+[Klq]+[K2qq]){q}  =  {F(t)}  (6) 


where  the  modal  mass,  damping,  and  linear  stiffness  matrices  are 

([m],  \c\  [k])  =  [OJ'T  ([M],  [C],  [K]  -  [Kot  ])[<I.]  (7) 


and  the  quadratic  and  cubic  terms  in  modal  coordinates  and  the  modal  force 
vector  are 


[Klq]{q}  =  [<E.f  2^qr[Kl]« 

lr=l 

[.K2qq]{q}  =  [3.]^  ^^q,q,[K2] 


[<E>]{q} 


Vr=l  s=l 


(8) 

(9) 


{F}  =  [cI>f({P(t)}+{PT})  (10) 

AH  modal  matrices  in  eq.  (6)  are  constant  matrices.  With  given  initial 
conditions,  the  response  of  modal  coordinates  {q}  can  be  determined  from  eq. 
(6)  with  any  direct  numerical  integration  scheme  such  as  the  Runge-Kutta  or 
Newmark-P  method.  Therefore,  no  updating  of  the  vibration  modes  is  needed. 
The  following  is  the  description  of  the  selection  of  the  initial  conditions  for 
periodic  motions. 

With  the  input  of  suitable  initial  conditions,  three  types  of  solutions,  periodic 
or  nearly  simple  harmonic,  superharmonic  and  subharmonic  solutions,  can  be 
obtained.  The  selection  of  each  type  solution  is  based  on  the  solution  of  the  1- 


1426 


DOF  Duffing  equation  obtained  by  using  the  modal  reduction  method 
described  earlier.  For  example,  the  system  equations  of  motion  of  a  symmetric 
composite  plate  can  be  reduced  to  1-DOF  model  (Note:  the  quadratic  term  is 
gone  because  the  plate  is  symmetric)  as 

Mjq,+Crqr+Krqr+K2,q^  =FrSin(C0t)  (11) 

where  Mr,  Cr,  Kr,  K2r  and  Fr  are  scalar  constants  and  represent  the  modal  mass, 
damping,  stiffness  and  force;  co  is  the  forcing  frequency  and  the  subscript  “r” 
denotes  the  linear  modal  number.  The  solution  of  eq.  (11)  can  be  assumed  as 
=AiCos(cot)+  A3cos(3cot)  for  the  simple  harmonic  and  superharmonic 
solutions,  then  two  sets  of  Ai  and  A3  can  be  obtained  by  the  substituting  of  the 
assumed  qr  into  eq.(n).  One  set  is  for  the  simple  harmonic  solution  and  the 
other  set  is  for  the  superharmonic  solution.  Based  on  these  solutions,  the  initial 
displacement  of  the  r-th  modal  coordinate  in  eq.  (6)  is  chosen  as  A1+A3.  The 
initial  velocities  and  all  other  initial  displacements  are  zero.  Similarly,  it  is 
assumed  that  qr  =Aicos(cot)+  Ai/3Cos(cot/3)  for  the  subharmonic  solution. 
Then,  all  those  initial  conditions  can  be  found  by  repeating  the  procedure  just 
described. 


Results  and  Discussions 

A  simply  supported  eight-layer  symmetrically  laminated  (0/45/-45/90)s 
composite  plate  is  studied  in  great  details.  The  plate  isofl5xl2x  0.048  in. 
(38.1  X  30.5  X  0.122  cm).  The  inplane  boundary  conditions  are  immovable,  i.e. 
u=v=0  on  all  four  edges.  The  graphite-epoxy  material  properties  are  :  Ei=22.5 
Msi  (155  GPa),  E2  =1.17  (8.07),  G12  =  0.66  (4.55),  V12  =  0.22  and  p  =  0.1458 
X  10'^  ib-sVin^  (1550  kg/m^).  The  C‘  conforming  rectangular  plate  element  is 
used  in  the  finite  element  model  and  the  plate  is  modeled  with  12  x  12  (144 
elements)  mesh.  The  element  has  a  total  of  24  DOF  (16  bending  and  8 
membrane).  The  lowest  six  natural  frequencies  (cOr,  r=l,6)  and  their 
corresponding  mode  shapes  are  :  coi  =  55.46  Hz  for  (1,1)  mode,  Oh  =  125.736 
Hz  for  (2,1)  mode,  CO3  =  151.951  Hz  for  (1,2)  mode,  CD4  =  216.475  Hz  for  (2,2) 
mode,  CO5  =250.585  Hz  for  (1,3)  mode  and  cOe  =  310.774  Hz  for  (3,1)  mode. 


1427 


Two  load  cases  considered  are  uniformly  distributed  harmonic  excitation  over 
the  plate  with  and  without  the  piresence  of  temperature.  A  constant  modal 
damping  factor,  =  C,/(2Mr  0)r),  of  0.02  and  a  four-mode  solution  are  used  in 
the  examples  (Only  the  (1,1),  (2,2),  (1,3)  and  (3,1)  modes  are  considered 
because  the  uniformly  distributed  excitation  cannot  induce  any  response  of  the 
(1,2)  and  (2,1)  modes,  see  Table  7  of  [1 1]). 

Harmonic  Excitations 

A  uniformly  distributed  pressure  load  of  the  form  p(x,y,t)  =po  sincot  is 
considered  for  the  forced  vibration  problem.  The  force  intensity  is  maintained 
at  po  =  0.00438  psi  (30.2  Pa),  however,  the  forcing  frequency  co  is  varying  in  a 
wide  range  from  0  to  4.5  times  of  the  lowest  linear  natural  frequency  ©i.  The 
results  are  shown  in  Figs.  1  to  4,  where  the  designations  for  the  total  responses 
are  indicated  in  Fig.  1 ,  while  the  time-histories,  phase  plots  and  power  spectra 
are  given  in  Figs.  2  to  4.  To  make  clear  the  behavior  of  the  vibration  response 
at  particular  frequency,  each  modal  coordinate  is  depicted  for  understanding 
the  simple  harmonic,  superharmonic  and  subharmonic  response  of  the  nonlinear 
system. 

Figures  2a-c  correspond  to  the  responses  of  the  three  forcing  frequencies  at 
0.6,  2.4  and  3.8  times  of  ©i  labeled  as  (1)  to  (3)  in  Fig.  1.  It  can  be  seen  that 
the  total  response  of  the  centre  of  the  plate  is  dominated  by  the  first  mode  (It 
should  be  noted  that  the  centre  of  the  plate  has  zero  contribution  from  the  (2,2) 
mode).  The  frequency  responses  of  the  four  modal  coordinates  are  composed 
of  superharmonic  frequency  components  of  order  2,  3,  5  ..  etc.  ,  as  well  as  the 
input  driving  force  frequency.  At  the  frequency  of  the  point  (3)  in  Figs.  2c  and 
2d,  the  corresponding  time  histories  of  the  second,  third  and  fourth  modal 
coordinates  are  unsymmetric  as  that  of  the  first  modal  coordinate  is  symmetric. 
Hence,  the  plate  is  vibrating  with  a  non-zero  equilibrium  position  due  to  the 
unsymmetric  responses  of  the  second,  third  and  fourth  modes. 

In  Figs.  3a  and  3b,  which  correspond  to  the  responses  of  two  points  (4)  and 
(5)  at  ©  =  2.4  ©1  and  4.2  ©i  in  Fig.  1,  the  total  response  of  the  centre  of  the 
plate  is  almost  pure  simple  harmonic  at  that  particular  frequency  range. 

In  Fig.  4a,  which  corresponds  to  the  response  of  the  point  (6)  at  ©  =  3.8  ©i 
in  Fig  1,  the  centre  of  the  plate  is  mainly  composed  of  subharmonic  response  of 
order  1/3.  In  the  time  histories  of  the  four  modal  coordinates  of  Fig.  4b,  it  can 
be  seen  that  the  subharmonic  component  in  the  total  response  is  contributed  by 


1428 


the  (1,1)  modal  coordinate,  and  the  responses  of  the  higher  modal  coordinates 
are  pesudo  harmonic. 

Combined  Harmonic  and  Thermal  Loads 

In  addition  to  the  uniform  pressure  po  sincot,  a  steady  state  temperature 
change  of  2.9Tcr  is  also  applied  to  the  composite  plate  (  where  the  buckling 
temperature  =13.79  °F).  The  forcing  frequency  is  taken  as  co  =  1.45c0i  and 
it  is  kept  at  that  constant  frequency  of  excitation,  however,  three  pressure 
intensities  at  po  =  6,  10  and  14  x  10*^  psi  are  considered.  The  responses  are 
shown  in  Figs.  5a-c.  The  results  are  shown  after  the  transient  response  being 
damped  out,  this  is  demonstrated  by  the  quasi-steady  state  time  histories  in 
Figs.  5a  and  5c. 

When  the  pressure  load  is  small  at  0.006  psi  (41.3  Pa),  the  plate  exhibits 
small  oscillations  about  one  of  the  thermally  buckled  positions  (Wmax/h 
=1.0237)  shown  in  Fig.  5a.  With  increase  of  the  pressure  loading,  the  amplitude 
of  vibration  increases.  Fig.  5b  shows  the  so-called  snap-through  or  oil-canning 
phenomenon  at  po  =  0.010  psi  (68.9  Pa),  the  plate  behavior  is  chaotic  and  has 
two  potential  weUs.  With  the  further  increase  of  the  pressure  loading,  the  plate 
exhibits  large  amplitude  oscillations  through  the  two  buckled  positions  as 
shown  in  Fig.  5c  at  po  =  0.014  psi  (96.4  Pa).  The  plate  motion  is  periodic  at 
low  and  high  pressure  loads,  however,  the  plate  response  composes  of 
superharmonic  frequency  components  of  order  2  and  3  at  the  low  pressure  and 
of  order  3  and  5  at  the  high  pressure  as  shown  in  the  PSD  plots. 

The  substance  of  the  transition  of  the  three  distinct  plate  behaviors,  from  the 
small  oscillations  into  the  chaotic  motion  then  into  the  large  amplitude 
vibrations  with  the  increase  of  forcing  intensity,  is  shown  in  Fig.  6.  In  the  low 
pressure  range,  the  plate  could  also  vibrate  about  the  another  equally  possible 
bifurcation  buckled  position  shown  with  dotted  lines. 


Conclusion 

Based  on  the  finite  element  method,  a  multimode  time-domain  formulation 
for  nonlinear  forced  vibration  of  composite  plates  is  presented.  The  main 
advantage  of  this  method  is  that  the  system  matrix  equation  derived  from  the 
FEM  can  be  transformed  into  a  set  of  general  type  Duffing  equations  with 
constant  system  matrices  and  much  smaller  DOF.  The  selections  of  initial 


1429 


conditions  for  the  subharmonic,  simple  harmonic  and  superharmonic  responses 
are  presented.  Through  detailed  descriptions,  the  frequency  response 
characteristics,  phase  plots,  time  histories  and  the  power  spectrums  have  been 
illustrated  for  the  three  types  of  responses.  The  responses  of  a  thermally 
buckled  composite  plate  under  harmonic  excitation  with  fixed  forcing 
frequency  and  various  amplitudes  are  also  obtained.  Snap-through  motion  is 
observed  at  moderate  pressure  loads. 


References 

1.  J.  M.  Whitney  and  A.  W.  Leissa  1969  Journal  of  Applied  Mechanics  36,  261-266. 
Analysis  of  heterogeneous  anisotropic  plates. 

2.  C.  Y.  Chia  1988  Applied  Mechanics  Review  41,  439-451.  Geometrically  nonlinear 
behavior  of  composite  plate:  A  review. 

3.  C.  Y.  Chia  1980  Nonlinear  Analysis  of  Plates,  McGraw-Hill,  New  York. 

4.  M.  Sathyamoorthy  1987  Applied  Mechanics  Review  40,  1553-1561.  Nonlinear 
vibration  analysis  of  plates:  A  review  and  survey  of  current  developments. 

5.  A.  H.  Nayfeh  and  D.  T.  Mook  1979  Nonlinear  Oscillations,  John  Wiley,  New 
York. 

6.  H.  F.  Wolfe,  C.  A.  Shroyer,  D.  L.  Brown  and  L.  W.  Simmons  1995  Technical 
Report  WL-TR-96-3057,  Wright  Laboratory,  Wright  Patterson  AFB,  Ohio.  An 
experimental  investigation  of  nonlinear  behavior  of  beams  and  plates  excited  to 
high  levels  of  dynamic  response. 

7.  J.  N.  Reddy  1985  Shock  and  Vibration  Digest  17,  3-8.  A  review  of  the  literature 
on  finite  element  modeling  of  laminated  composite  plates. 

8.  Y.  Shi  and  C.  Mei  1996  Proceedings  of  the  37th  AIAA  /  ASME  /  ASCE  /  AHS  / 
ASC  Structures,  Structural  Dynamics,  and  Material  Conference,  Salt  Lake  City, 
UT,  1355-1362.  Coexisting  thermal  postbuckling  of  composite  plates  with  initial 
imperfections  using  finite  element  modal  method. 

9.  C.  K.  Chiang,  C.  Mei  and  C.  E.  Gray,  Jr.  1991  Journal  of  Vibration  and  Acoustics 
113,  309-315.  Finite  element  large-amplitude  free  and  forced  vibrations  of 
rectangular  thin  composite  plates. 

10.  A  K.  Noor  1981  Composites  and  Structures,  13,  31-44.  Recent  advances  in 
reduction  methods  for  nonlinear  problems. 

11.  Y.  Y.  Lee,  Y.  Shi  and  C.  Mei  1997  Proceedings  of  the  6th  International 
Conference  on  Recent  Advances  in  Structural  Dynamics,  University  of 
Southampton,  UK.  A  finite  element  time  domain  multi-mode  method  for  large 
amplitude  free  vibration  of  composite  plates. 


1430 


Centre  Disp/Thlckness  Centre  Disp/Thickness 


Po=0.00438  Psi 
Damp.  Ratio  =  0.02 


(3) 


(2) 


(1) 


■  ■ 


(4) 


(6) 


(5) 


0  1  2  oa/osl  3  4  5 

Freq.  Ratio 

Figure  1.  Frequency  response  of  the  simply  supported  (0/45/-45/90)s  rectangular  plate 


2.5 
2 

1.5 
1 

0.5 
0 

-0.5 
-1 
-1.5 
-2 
-2.5 

0  0.002  0.004  0.006  0.008  0.01  0.012  0.014  0.016  0.018  0.02 

Po  (psi) 

Figure  6.  Plate  centre  response  vs  force  amplitude  at  co  =  1.45ci>i  and  T  =  2.9  Ter 


1431 


1432 


1433 


1434 


1435 


1436 


GEOMETRICALLY  NONLINEAR  RESPONSE  ANALYSIS 
OF  LAMINATED  COMPOSITE  PLATES  AND  SHELLS 


C.W.S.  Tot  and  B.  Wangt 

Department  of  Mechanical  Engineering 
University  of  Nebraska 
255  Walter  Scott  Engineering  Center 
Lincoln,  Nebraska  68588-0656 
U.S.A. 

E-mail;  cwsto@unlinfo.unLedu 


Abstract 


The  investigation  reported  in  this  presentation  is  concerned  with  the  prediction 
of  geometrically  large  nonlinear  responses  of  laminated  composite  plate  and 
shell  structures  under  dynamic  loads  by  employing  the  hybrid  strain  based  flat 
triangular  laminated  composite  shell  finite  elements.  Large  deformation  of 
finite  strain  and  finite  rotation  are  emphasized.  The  finite  element  has  eighteen 
degrees  of  freedom  which  encompass  the  important  drilling  degree  of  freedom 
at  every  node.  It  is  hinged  on  the  first  order  shear  deformable  lamination 
theory.  Various  typical  laminated  composite  plate  and  shell  structures  under 
dynamic  loads  have  been  studied  and  representative  ones  are  presented  and 
discussed  in  this  paper.  Shear  locking  has  not  appeared  and  there  is  no  zero 
energy  mode  detected  in  the  problems  studied.  It  is  very  accurate  and  efficient. 
Consequently,  it  is  relatively  much  more  attractive  than  other  elements 
currently  available  in  the  literature  for  large  scale  nonlinear  dynamic  response 
analysis  of  laminated  composite  plate  and  shell  structures. 


t  Professor  and  corresponding  author 
i  Research  Associate 


1437 


1.  INTRODUCTION 


Many  modem  structures  such  as  nuclear  reactor  containment  installations, 
naval  and  aerospace  structures,  and  their  components,  must  be  designed  to 
withstand  a  variety  of  intensive  dynamic  disturbances.  Because  of  their  many 
attractive  features  over  isotropic  materials  more  and  more  stmctures  or 
components  in  the  aforementioned  systems  are  made  of  laminated  composite 
materials.  The  investigation  reported  in  this  paper  is  therefore  concerned  with 
the  prediction  of  geometrically  large  nonlinear  responses  of  laminated 
composite  plate  and  shell  structures,  of  complicated  geometries,  under  transient 
excitations.  With  complicated  geometries  analytical  solution  is  impossible  and 
therefore  a  versatile  numerical  method,  the  finite  element  method  has  been 
employed.  A  hybrid  strain  based  flat  triangular  laminated  composite  shell 
finite  element  has  been  developed  by  the  authors  [1,2]  for  the  nonlinear 
analysis  of  plate  and  shell  structures  under  static  loadings.  The  present 
investigation  is  an  extension  of  [1,2]  to  cases  with  the  aforementioned 
dynamic  forces.  Among  various  attractive  features  of  the  derived  element 
stiffness  and  consistent  element  mass  matrices  five  are  worthy  of  listing  here 
for  completeness.  These  are:  (a)  their  ability  to  deal  with  large  nonlinear 
elastic  response  of  finite  strain  and  finite  rotation,  (b)  the  fact  that  they  are  in 
explicit  expressions  and  therefore  no  numerical  integration  is  necessary,  (c)  the 
obtained  results  of  a  relatively  comprehensive  tests  [2,  3]  show  that  the 
element  is  free  from  shear  locking,  (d)  the  element  gives  correctly  six  rigid 
body  modes,  and  (e)  the  finite  element  has  three  nodes  and  eighteen  degrees 
of  freedom  (dof)  which  encompass  the  important  drilling  degree  of  freedom 
(ddof)  at  every  node.  It  is  based  on  the  first  order  shear  deformable  lamination 
theory.  It  is  a  generalization  of  the  low-order  flat  triangular  shell  element  for 
isotropic  materials  developed  earlier  by  Liu  and  To  [4]. 

It  is  noted  that  one  of  the  earlier  work  that  employed  triangular  shell 
element  is  due  to  Noor  and  Mathers  [5].  In  the  latter  a  mixed  type  triangular 
element  was  proposed.  The  element  has  six  nodes,  and  78  dof.  It  was  based 
on  the  shallow  shell  theory  and  was  shear  deformable.  Recently,  Lin  et  al  [6] 
developed  a  finite  element  procedure  to  analyze  composite  bridges.  The  finite 
element  procedure  was  based  on  small  elasto-plastic  strains  and  updated 
Lagrangian  formulation.  The  element  used  was  flat  and  constructed  by  the 
superposition  of  a  discrete  Kirchhoff  bending  element  and  a  linear  strain 
triangular  membrane  element.  It  has  six  nodes.  There  are  three  translational 
and  three  rotational  dof  at  its  comer  nodes  and  three  translational  dof  at  mid¬ 
side  nodes.  In  1994  a  flat  triangular  shell  element  was  presented  for  static 
nonlinear  analysis  by  Madenci  and  Barut  [7].  It  is  based  on  the  so-called  free 
formulation  concept  for  analyzing  geometrically  nonlinear  thin  composite 
shells.  A  corotation  form  of  the  updated  Lagrangian  formulation  is  utilized. 
The  theoretical  basis  was  on  the  geometrically  nonlinear  Kirchhoff  plate  theory 
without  considering  the  effects  of  transverse  shear  deformation.  The  element 


1438 


is  of  displacement  type.  It  has  three  nodes  and  six  dof  for  each  node.  While 
such  formulation  has  some  advantageous  features  computationally  the  element 
is  relatively  less  efficient  because  (a)  the  linear  element  stiffness  matrix 
consists  of  a  basic  and  a  higher-order  stiffness  matrices  in  the  sense  of 
Bergan  and  Nygard  [8]  for  isotropic  materials,  and  (b)  the  important  effects 
of  transverse  shear  deformation  in  the  plate  component  of  of  this  element  has 
been  disregarded.  A  more  recent  contribution  on  triangular  elements  is  made 
by  Zhu  [9].  The  natural  approach  is  used  to  construct  a  curved  triangular  shell 
element  for  static  analysis  of  geometrically  nonlinear  sandwich  and  composite 
shell  structures.  The  element  has  six  nodes.  There  are  six  dof  at  each  corner 
node  and  three  dof  at  each  mid-side  node.  Updated  Lagrangian  description  was 
adopted  in  the  procedure.  In  the  element  formulation  the  transverse  shear 
deformation  was  considered  by  assuming  constant  transverse  shear  stress 
distribution. 

In  the  next  section  the  formulation  of  element  stiffness  matrices  is 
outlined.  Section  3  deals  with  the  derivation  of  element  matrices.  Section  4  is 
concerned  with  the  application  of  the  derived  elements  to  three  example 
problems  of  plate  and  shell  structures.  The  concluding  remarks  are  included 
in  Section  5. 


2.  FORMULATION  OF  ELEMENT  STIFFNESS  MATRICES 

Finite  element  formulation  for  the  derivation  of  a  family  of  simple 
three-node,  six  dof  per  node,  hybrid  strain  based  laminated  composite 
triangular  shell  finite  elements  for  large  scale  geometrically  nonlinear  analysis 
is  briefly  outlined  in  this  section.  Large  deflection  of  finite  strains  and  finite 
rotations  are  included.  The  first  order  shear  deformation  theory  and  the 
degenerated  three  dimensional  solid  concept  are  adopted.  In  particular,  element 
matrices  for  one  member  of  the  family  are  derived  explicitly  with  the 
symbolic  computer  algebra  package  MACSYMA.  To  minimize  the  algebraic 
manipulation  involved  in  the  derivation,  updated  Lagrangian  description  is 
employed  in  the  incremental  formulation  of  the  finite  element  procedure.  In 
essence,  the  present  formulation  is  an  extension  of  the  work  by  Liu  and  To  [4] 
for  isotropic  materials  to  multi-layer  laminated  composite  shells.  Therefore,  in 
the  development  the  present  approach  follows  closely  that  of  the  last  reference. 

2.1  Incremental  variational  principle 

The  Hellinger-Reissner  functional  tChr  can  be  written  as 


where, 

e'  is  the  independently  assumed  strain  field; 

e“  is  the  strain  due  to  displacement; 

C  is  the  material  stiffness  matrix  or  elasticity  matrix; 

W  is  the  work  done  by  external  forces, 
and  the  superscripts  e  and  u  indicate  that  the  quantities  are  from  independently 
assumed  strain  field  and  displacement  field,  respectively.  For  geometrically 
nonlinear  analysis  with  incremental  formulation  and  updated  Lagrangian 
description,  the  static  and  kinematic  variables  in  current  equilibrium 
configuration  at  time  t  are  assumed  to  be  known  quantities  and  the  objective 
is  to  determine  their  values  in  the  unknown  subsequent  equilibrium 
configuration  at  time  t+At.  For  a  time  increment  At,  that  is  from  time  t  to 
(t+At),  one  has 

-  ATTjjj^CAUjAe®)  -  7ifjjj(t+At)  -  ttHR®  , 

or,  with  reference  to  equation  (1), 

Atihr  "  /  [(e')TC(Ae“) 

-  i(Ae‘)fC(Ae')  - 

where, 

Au  is  the  vector  of  incremental  displacement; 

Ae*  is  the  vector  of  independently  assumed  incremental  updated 
Green  strains; 

Ae“  is  the  vector  of  incremental  updated  Green  ’geometric’  strains 
or  incremental  Washizu  strains; 

AW  is  the  work-equivalent  term  corresponding  to  prescribed  body 
forces  and  surface  tractions  in  configuration  C 

Equation  (3)  is  the  incremental  form  of  Hellinger-Reissner  variational 
principle.  For  updated  Lagrangian  description,  the  integral  is  evaluated  at  the 
current  configuration  C  In  the  equation,  the  term 

J(Ae')^C(e^  -  e“)  dV 

Ve 


■f  (Ae'')'^C(Ae“) 


(3) 


(Ae')'^C(e‘^  -  e'^)]  dV  -  AW  . 


1440 


is  the  so-called  compatibility-mismatch.  Numerical  results  of  Saleeb  et  al  [10] 
showed  that  though  totally  discarding  the  term  resulted  in  convergence 
difficulties,  while  including  the  term  in  only  the  first  iteration  of  every  load 
step  yielded  essentially  the  same  results  as  those  having  the  term  under  all 
circumstances.  However,  Liu  and  To  [4]  reported  no  difficulties  for 
convergence  when  the  term  was  ignored.  In  the  current  study,  this  term  is  also 
disregarded.  Then  equation  (3)  can  be  recasted  as 


^^hr“/  [  +  (Ae®)^C(Ae“) 

Vc 


(4) 


-■i(Ae')’'C(Ae')]dV-AW, 

2 


where  =  (e^)^C  is  the  Cauchy  (true)  stress  vector  at  the  current 
configuration  C  ^  In  this  equation,  the  incremental  Washizu  strain  Ae“  can  be 
expressed  in  two  parts 


Ae^ 


Aei"  ^  ATii“ 


(5a) 


and  they  are  related  to  the  incremental  displacement  by 


|(Auy  AUjj)  .  Arii” 


-AUyAU,^ 


(5b, c) 


where  the  Einstein  summation  convention  for  indices  has  been  adopted  and  the 
differentiation  is  with  respect  to  reference  co-ordinates  at  the  current 
configuration  C 

Substituting  equation  (5a)  into  (4)  yields 

A’'™  =  f  [-  -(Ae')’'C(Ae')  +  (Ae')'^C(A£")  +  a'^Ae” 

V.  2  (6) 

+  o^Aii"  +  (Ae')fCATi“  ]  dV  -  AW  , 


where  a  is  the  Cauchy  stress  vector. 

Discarding  the  higher  order  term,  )  ^Aq  ^  results  in 

AjIhr  “  /  [-  -(Ae')'^C(Ae')  +  (Ae')TC(AE“) 

+  a^Ae'^  +  a^Aq^]  dV  -  AW  . 


(7) 


1441 


2.2  Hybrid  Strain  Formulation 

Element  stiffness  matrices  for  a  hybrid  strain  based  finite  element  can 
be  derived  directly  from  equation  (7).  Generally  the  independently  assumed 
strain  field  and  displacement  field  can  be  written  as 
Ae'  =  P  Att  ,  Au  =  <t)  Aq 


where  P  is  the  strain  distribution  matrix,  (|)  is  the  displacement  shape  function 
matrix,  Aa  is  the  vector  of  incremental  strain  parameters  and  Aq  is  the 
incremental  nodal  displacement.  Substituting  equations  (8a,  b)  into  (7),  and 
defining 

H  -  J  P^CP  dV^  ,  G,  =  /  P'^CBl  dV^  , 

V  -  /  dVe  .  F.  "  /  BlO  dV.  , 

V  V, 


one  can  show  that 

AT:jnj(Aq,Aa)  =  ’  ^Aa'^HAa  +  Aa'^G^Aq 

+  F^Aq  +  ■iAq^'k^LAq  -  F'^Aq  ] 


(10) 


where  F  is  the  external  nodal  force  vector  in  the  neighbour  configuration 
associated  with  the  AW  term  in  equation  (7);  and  B^l  are  the  linear  and 
nonlinear  strain-displacement  matrices,  while  Cq  is  the  matrix  containing  the 
Cauchy  stress  components  at  the  current  configuration. 

Finally,  one  can  show  that 

(k[  .  V)  Aq  -  F{t .  At)  -  F,  ,  k[  =  g7h-‘G.  0 


where  the  expression  in  equation  (1  lb)  is  the  element  "linear"  stiffness  matrix. 
The  term  k^L  defined  in  equation  (9)  is  the  "nonlinear"  or  initial  stress 
stiffness  matrix  and  Fi  is  the  pseudo-force  vector.  The  right  hand  side  of 
equation  (11a)  is  the  equilibrium  imbalance. 


1442 


3.  ELEMENT  MATRICES  AND  THEIR  UPDATING 


The  derivation  of  nonlinear  element  stiffness  matrices,  constitutive 
equations,  mass  matrices  of  the  element  shown  in  Figure  1  are  outlined  here. 
In  addition,  updating  of  configuration  and  stresses  at  every  time  step  is 
considered  here  for  completeness. 


3.1  Nonlinear  Element  Stiffness  Matrices 

For  the  assumed  displacement  field,  an  arbitrary  point  within  an 
element  is  governed  by 


r*' 

3 

t 

h 

t 

Si 

0 

3 


(12) 


The  incremental  displacements  of  an  arbitrary  point  within  the  element  are 


Au'  1 

1  O 

Au/| 

3 

< 

II 

AVi  1 

>  +  (AVi')  . 

i-l 

Aw'  , 

1 

Aw/ 

1*1 

Employing  quadratic  polynomials  for  the  translational  dof  and  including  ddof 
lead  to 


Au'  ^ 

3 

AUi' 

3 

Ai(ii)  Ai(i2) 

A0' 

Av' 

Aw' 

=  E?i^ 

i-l 

AVi' 

Awj' 

^  c'E^i 

i-l 

Ai(21)  Ai(22) 

Ai(31)  Ai(32)  ^ 

1-if 

0 

0 

Pi 

AO^i 

0 

0 

5i 

A 

Aei  • 

.  "Pi 

-5i 

0 

Ae' 

(14) 


1443 


In  the  foregoing, 

U  =  Ui^l  +  U2^2‘^U3^3  +  Piejl  +  p20t2-"P3®G 

W  =  WiCl+W2^2-"W3^3-Pie,l-P2e  ^-536,3 

"5i®s1“^2®s2“^3®s3  » 

0r  =  0rl^l'*'®i2^2-*'®r3^3 
®s*®sl^l  ■*'®s2^2'^®s3^3 
®tl^l  '*’^12^2  ®e^3 


(15a) 


(15b) 


where 


Pi  =■  (^1^3  "^12^2)^!  ’  5i  “  (^31^3  “^12^2)^!  » 

P2  *  (^12^1  ”^3^3) ^2  »  ^2  *  (^12^1  “^23 ^3) ^2  ’ 

P3  “  (^3^2  ~  ^31  ^1)^3  »  ^3  ”  (^23^2  1)^3  • 

The  remaining  symbols  have  been  defined  by  Liu  and  To  [4]  and  are  not 

repeated  here  for  brevity. 

For  the  assumed  strain  field,  the  strain  vector  in  equation  (8a)  may  be 
written  as 


Ae*^ 


Ae; 

4 

Ay' 


(16) 


where 

A^'m  =  PmA«„  .  Ax'  -  PfcAa, 

Ay'  -  PsA«s 


(17a) 


1444 


with 


Aa^-lAttj  Aa2  Actj)’'’  , 

(17b) 


Aa^-{Aa4Aa5Aa^}'^,Aa^-{Aa7Aa8Aa,}’', 


and 


1 

0 

o' 

1 

0 

o' 

B 

II 

0 

1 

0 

11 

pf 

0 

1 

0 

o, 

0 

1. 

0 

0 

1 

-83(1-2^2)  83(1-2^1)  0 

-13(1-2^2)  (r3-r2)(l-2^i)  12(1-2^3) 


where  the  subscripts  m,  b  and  s  denote  the  membrane,  bending  and  transverse 
shear  components  of  P  in  equation  (8). 

By  defining 

=  /  PjA'P^  da  ,  =  f  P3"C;P„  da  , 

a  a 

-  /  Pjc^P.  da  ,  -  /  Ps^E'P,  da  , 

a  a 

=  /  PjB'P^  da  ,  =  /  P,"CJP,  da  ,  (18) 

a  a 

H„b  -  /  PmB'Pb  da  ,  P,^CX  da  . 

a  a 

Hbb  =  /  Pb'D'Pb  da 


1445 


where  a  is  the  area  of  the  triangular  shell  element  and 


A'  -  E  (C.)k(h,-h,.,) .  B'  -  i  ^  (C,)k(h^h,l,) , 

k-1  ^  k-1 

"  'T  S  (^a)k(^k“'^k-l)  ’  (^b)k(^k“^k-l)  ’ 

d  k-1  k-1 

Ca  -  i:  (c  j,aik“Vi)  >  Cb  -  -i  i:  (c 

k-1  k-1 


(19) 


in  which  the  integer  n  is  the  number  of  laminae  in  the  laminated  composite 
structure.  Then  the  matrix  H  in  equation  (9)  becomes 


H  = 

Hta 

Hu 

H* 

9x9 

(20) 


Similarly  by  defining 


/  pJa'B^  da  . 

Gsm  -  /  Ps"cX  <ia 

a 

a 

Gms  ■ 

/  da  , 

=  /  Ps^E'B,  da  , 

a 

a 

Gbm  = 

/  P,^B'B„  da  , 

Gi,  ■=  /  pXb,  da  . 

a 

a 

Gmb  “ 

/  pJb'B,  da  , 

G.,  -  /  PsXX  ^  . 

a  a 


G,,  =  /  pJd'B,  da 


1446 


one  has  the  matrix 


+ 

G„b 

+ 

G^ 

B 

X) 

o 

+ 

Gbb 

+ 

Gbs 

Gs. 

+ 

G.b 

+ 

9x18 

(22) 


Therefore,  with  the  ddof  considered  the  element  stiffness  matrix  can  be  shown 
to  be 


k  -  k[ 


+  k^  +  k^L 


(23) 


where  the  linear  element  stiffness  matrix  k^’  and  the  "nonlinear"  or  initial 
stress  stiffness  matrix  k^L  ^re  defined  by  in  equations  (9)  and  (11),  while  the 
stiffness  matrix  associated  with  the  ddof  k^d  is  defined  as 


kda  =  i:(G.),(h,A-.)/B7B,da, 


(24) 


k-1 


in  which 

Bd  =  [Bdi  Bj2  B^Ji^ig 


and 


Bdi 


-Ui.r  0  0  0 


with  i  =  1,2,3. 

The  "nonlinear"  or  initial  stress  stiffness  matrix  k^L  can  be  obtained  if 
the  nonlinear  strain-displacement  matrix  B^l  and  the  matrix  Gc  which  contains 
the  Cauchy  stress  components  at  the  current  configuration  are  available.  The 
matrix  B^l  is  defined  by  equation  (45)  of  Liu  and  To  [4]. 

The  matrix  Oq  is  constructed  from  the  Cauchy  stress  vector  c  and 
defined  as 


^11^ 

°3ll3 

^22^3 

”23'^ 

*^23^ 

O3 

(25) 


1447 


with  I3  being  the  3x3  identity  matrix  and  O3  a  3x3  null  matrix.  The  transverse 
stress  components  of  a  are  considered  constant  over  the  thickness,  and  all 
components  of  a  are  calculated  and  updated  for  each  time  step  at  the  centroid 
of  each  element. 

3.2  Constitutive  Equations 

For  finite  strain  problems  in  the  elastic  range,  the  reduced  stiffness 
matrix  is  a  function  of  stresses.  To  incorporate  finite  strains  in  the  analysis, 
several  approaches  can  be  applied.  The  following  adopted  from  reference  [4] 
is  to  add  the  linear  elastic  matrix  a  correction  matrix  which  is  a  function  of 
Cauchy  stress.  To  begin  with,  the  correction  terms  in  tensor  form  becomes 

Ciid  "  -  *  OjiSfl  +  OnSik  +  "jiSik  ) 


where  8^^  is  the  Kronecker  delta.  Note  that  this  equation  comes  as  a  result  of 
transforming  the  Jaumann  stress  rate  to  the  incremental  second  Piola-Kirchhoff 
stress.  If  the  stress  and  strain  vectors  are 

O  =  {  Ojj  O22  O33  0^2  ^23  Ojj  }  , 

e  =  {  ®22  ®33  ®12  ®23  ®31 


the  matrix  form  of  equation  (26)  is 


0 

0 

2012 

0 

2013 

0 

4<J22 

0 

20i2 

20,3 

0 

1 

0 

0 

4033 

0 

2023 

2013 

2 

20i2 

2a,2 

0 

^ll'^^22 

Oi3 

^23 

0 

2°23 

2023 

°13 

^22"^  ^33 

^12 

2oj3 

0 

20j3 

^^23 

^12 

Oii  +  O 

In  present  investigation  the  so-called  degenerated  concept  is  adopted  and 
therefore  the  elastic  modulus  in  the  normal  direction  to  the  plane  of  the  shell 
structures  is  considered  zero.  Consequently  the  stress  and  strain  in  the 
transversal  direction  are  ignored.  In  the  linear  analysis  the  constitutive 
relations  for  a  lamina  have  been  defined  as 

o  =  Qe  (29) 


1448 


where 


°n  °  zx  ’ 

e  -  {  e,  By  e„  )■" 


Qn 

Qi2 

Qi6 

0 

0 

Qi2 

Q22 

Q26 

0 

0 

Qi6 

Q26 

0 

0 

0 

0 

0 

Q44 

0 

0 

0 

^5 

Q55 

The  corresponding  matrix  from  equation  (28)  is 


0 

2<’.y 

0 

2°. 

0 

40y 

2<^.y 

2o 

yz 

0 

2% 

°yy 

0 

2^. 

®  zx 

°xy 

0 

^xy 

The  material  stiffness  matrix  for  a  lamina  thus  becomes 

C  =  Q  +  C 


(30) 


(31) 


(32) 


(33) 


where 


t^aJ3x2 
t^aj2x3  J2x2 . 


in  which  C„  C^,  and  =  ^ba  are  given  in  equation  (19). 

With  the  consideration  of  large  deformation  and  finite  strain,  the 
constitutive  equations  for  a  multilayered  structure  or  laminate  can  be  written 
as 


1449 


or  simply 


(34) 


'  N  ' 

■  A' 

B' 

Ca 

^  M 

D' 

Cb 

- 

X 

Qs 

Ub 

e' 

8x8 

.  y  . 

^  ®N 

where  N,  M  and  Qj  are  the  vectors  of  stress  resultants  corresponding  to 
membrane,  bending  and  transverse  shear,  respectively.  The  matrices  A’,  B’, 
D’,  E’,  and  Cg  have  been  defined  in  equation  (19). 

3.3  Element  Mass  Matrices  and  Updating  of  Configurations  and 
Stresses 

In  the  present  study,  with  the  updated  Lagrangian  description,  the 
consistent  mass  matrix  is  formulated  in  the  current  configuration  C '.  The  mass 
matrix  is  then  updated  at  each  time  step.  The  assumptions  are  that  the  angular 
velocities  and  accelerations  are  small  enough  to  be  discarded.  By  following  the 
procedures  of  Liu  and  To  [4]  the  consistent  element  mass  matrix  can  be 
obtained  as 


in  which  mt„  and  m„t  are  translational  and  rotational  components  of  the 
consistent  element  mass  matrix,  respectively.  Matrix  m^  is  the  part  associated 
with  the  ddof.  When  it  is  used  for  the  incremental  formulation  with  updated 
Lagrangian  description,  updating  relevant  quantities  at  each  incremental  step 
are  required  before  evaluating  the  mass  matrix.  All  these  mass  matrices  are 
obtained  explicitly  with  the  symbolic  computer  algebra  package  MACSYMA. 

For  each  incremental  step,  the  configuration  and  stresses  have  to  be 
updated.  Details  of  the  steps  can  be  found  in  the  reference  by  Liu  and  To  [4] 
and  therefore  are  not  included  here.  However,  it  may  be  appropriate  to  point 
out  that  the  linear  consistent  element  matrix  for  multi-layer  composites  has 
been  employed  by  the  authors  [11]  for  vibration  analysis  of  plates  and  shells. 


4.  EXAMPLES  OF  LAMINATED  COMPOSITE  PLATE  AND  SHELL 

There  are  two  main  objectives  in  this  section.  First,  accuracy  of  results 
obtained  by  the  presently  derived  element  matrices  is  studied.  Second,  the 
validity  and  conceptual  adequacy  of  the  formulation  and  assumptions  made  in 


the  derivation  of  element  matrices  are  assessed.  For  brevity,  one  multi-layer 
plate,  one  multi-layer  shell  structure,  and  a  cantilever  panel  with  free  end  step 
moment  are  included  here.  More  example  problems  can  be  found  in  To  and 
Wang  [3],  and  Wang  and  To  [12]. 

4.1  Multi-Layer  Plate  Under  Uniformly  Distributed  Step  Disturbance 

The  square  plate  considered  has  two  layers.  Its  geometrical  dimensions 
are:  side  length  a  =  2.438  m  and  total  thickness  h  =  0.00635  m.  Each  layer  of 
the  laminate  has  equal  thickness.  The  plate  stacking  scheme^ is  cross-ply 
(0/90).  The  layer  material  properties  are:  ^  =  6.8974  x  N/m^,  Ej  =  25 
Gi2  =  G,3  =  0.5  E2,  G23  =  0.2  E2,  V12  =  0.25  and  density  p  =  2498.61  kg/m  . 

It  is  supported  by  hinges  at  its  four  edges.  At  these  edges  U  or  V  (note 
henceforth  upper  case  of  deformation  variable  refers  to  global  co-ordinate) 
parallel  to  the  edges  are  not  constrained.  These  boundary  conditions  are 
denoted  as  BCl  in  reference  [13].  For  the  purpose  of  direct  comparison  with 
the  results  reported  in  the  latter  reference,  one  quarter  of  the  plate  is  modeled 
by  a  4  X  4  D  mesh  (see  Figure  1  for  the  definition  of  D  mesh).  Thus,  the 
boundary  conditions  applied  are:  V  =  0^  =  0.0  at  AB,  U  =  W  =  0,,  =  0.0  at 
BC,  V  =  W  =  0y  =  0.0  at  CD  and  U  =  0y  =  0.0  at  AD.  In  addition,  all  0,  are 
constrained.  After  application  of  the  boundary  conditions  there  are  158 
unknowns  in  this  case. 

The  uniformly  distributed  transversal  step  disturbance  with  intensity  po 
=  490.5  N/m^  is  applied  to  the  plate.  In  the  analysis,  the  option  of  inclusion 
of  directors  [3,  12]  and  small  strain  are  selected.  The  time  step  size  is  At  = 
0.001  seconds.  The  responses  at  the  centroid  obtained  by  using  the  HLCTS 
element  are  plotted  in  Figure  2.  They  are  compared  with  those  reported  by 
Reddy  [13]  in  which  results  were  obtained  with  a  nine-node  rectangular 
isoparametric  element.  In  the  latter  transverse  shear  was  considered.  Excellent 
agreement  can  be  observed.  Before  leaving  this  subsection  it  may  be 
appropriate  to  mention  that  the  nonlinear  element  stiffness  matrix  presented 
in  reference  [13]  is  nonsymmetric  while  the  one  derived  in  the  present 
investigation  is  symmetric.  In  fact,  when  the  system  is  conservative  the 
nonlinear  element  stiffness  matrix  can  be  shown  to  be  symmetric. 

4.2  Spherical  Shell  Segment  Under  A  Uniformly  Distributed  Step 

Disturbance 

The  geometry  of  the  spherical  shell  is  shown  in  Figure  3  in  which  the 
shell  is  simply  supported.  The  geometrical  properties  are:  radius  R  =  10.0  m, 
the  side  length  of  the  projected  plane  b  =  0.9996  m  and  the  total  thickness  h 
=  0.01  m.  The  spherical  shell  is  considered  having  two  equal  thickness  layers 
and  they  have  the  (-  45/45)  lamination  scheme.  The  pertinent  material 
properties  are:  Ej  —  2.5x10^'  N/m^  E2  =  1.0x10^°  N/m^,  Gi2  =  G13  =  0.5x10  ^ 
N/m^,  G23  =  0.2x10*°  N/m^,  Poisson’s  ratio  V12  =  0.25  and  density  p  =  1.0x10 
kg/m^.  For  comparison  to  results  available  in  the  literature  one  quarter  of  the 


1451 


shell  is  modeled  by  the  proposed  hybrid  strain  based  shell  element  (identified 
as  HLCTS  for  brevity  and  convenience)  with  4  x  4  D  mesh.  The  boundary 
conditions  applied  to  the  finite  element  model  are:  V  =  =  0.0  at  line 

AB,V  =  W  =  ©,  =  0.0atBC,U  =  W  =  ©y  =  0.0  at  DC  and  U  =  ©y  =  ©,  =  0.0 
at  AD.  The  number  of  equations  to  be  solved  after  the  application  of  the 
boundary  conditions  is  189.  A  distributed  step  pressure  is  applied  to  its  outer 
surface  (pointing  toward  the  outer  surface).  It  has  an  intensity  p  =  2000.0 
N/ml  The  time  step  used  were  0.03  s,  0.01  s  and  0.005  s.  As  there  was  no 
significant  difference  and  for  efficient  reason  throughout  the  computation  the 
time  step  of  0.03  s  was  adopted.  The  nonlinear  transient  response  at  the  apex 
(central  point  A  of  the  shell)  is  obtained  and  plotted  in  Figure  4.  The  problem 
has  been  solved  by  Wu  et  al.  [14]  who  applied  a  curved  high-order 
quadrilateral  shell  element.  The  latter  has  48  dof  and  was  developed  based  on 
the  classical  lamination  theory.  It  is  observed  that  there  is  a  discrepancy  of 
about  8%,  with  respect  to  the  HLCTS  element  results,  for  the  amplitudes 
between  the  two  set  of  results.  However,  they  have  the  same  vibration  period. 
It  is  believed  that  the  present  results  are  more  accurate  as  the  element  used  in 
the  present  investigation  is  shear  deformable. 

4.3  Cantilever  Panel  With  Free  End  Step  Moment 

To  demonstrate  the  use  of  the  proposed  shell  element  for  structures 
undergoing  large  rotation  and  large  deformation  a  four  layer  cross-ply 
cantilever  panel  is  considered  here.  More  computed  results  for  this  case  can 
be  found  in  references  [3]  and  [12].  It  is  symmetrically  laminated  with  the 
stacking  scheme  (0/90/90/0).  Its  geometrical  properties  are:  L  =  1.2  m,  b  =  0.1 
m  and  h  =  0.01  m.  The  material  used  for  this  cantilever  is  the  high  modulus 
graphite/epoxy  composite.  Its  properties  are:  Ej  =  2.0685x10"  N/m^,  E2  = 
5.1713x10''  N/m^  G,-,  =  3.1028x10^  N/m^  G,,  =  G23  =  2.5856x10^  N/m^  p  = 
1605  kg/m^  and  Poisson’s  ratio  Vj2  =  0.25.  A  step  moment  M  about  an  axis 
parallel  to  the  width  of  the  panel  is  applied  to  the  free  end.  The  amplitude  of 
this  moment  is  =  1000.00  N-m. 

The  panel  is  discretized  by  a  12  x  1  A  mesh.  At  the  fixed  end,  all  dof 
are  constrained.  The  finite  element  model  has  144  unknowns. 

The  time  step  At  =  0.001  s  is  employed  in  the  trapezoidal  rule  direct 
integration.  The  nonlinear  transient  responses  at  the  end  of  the  cantilever  are 
solved  by  selecting  the  options  of  director  included,  small  strain  and  constant 
thickness  in  the  digital  computer  program  developed.  The  computed  end 
deflections  are  plotted  in  Figure  5.  As  noted  in  reference  [3,12],  the  inclusion 
of  directors  in  the  formulation  [15]  is  crucial  as  the  directors  are  important 
parameters  that  constitute  the  so-called  "exact  geometry"  for  large  rotation 
problems. 


1452 


5.  CONCLUDING  REMARKS 


The  hybrid  strain  based  laminated  composite  flat  triangular  shell 
(HLCTS)  element  for  the  static  analysis  of  geometrically  nonlinear  laminated 
composite  plates  and  shells  has  been  further  developed  and  employed  to  solve 
various  dynamic  problems.  A  relatively  comprehensive  study  for  various  plate 
and  shell  structures  idealized  by  this  element  has  been  performed  and  three 
representative  examples  are  included  to  demonstrate  its  accuracy,  efficiency 
and  conceptual  adequacy.  It  is  concluded  that  the  HLCTS  element  is  attractive 
for  large  scale  finite  element  analysis  and  modelling  of  shell  structures 
undergoing  geometrically  large  deformation  of  finite  strain  and  finite  rotations. 


ACKNOWLEDGMENT 

The  first  author  gratefully  acknowledges  the  financial  support  in  the 
form  of  a  research  grant  from  the  Natural  Sciences  and  Engineering  Research 
Council  of  Canada.  The  results  reported  above  were  obtained  in  the  course  of 
the  research  while  the  authors  were  at  the  University  of  Western  Ontario. 


REFERENCES 

1.  To,  C.W.S.  and  Wang,  B.,  Nonlinear  theory  and  incremental 
formulation  of  hybrid  strain  based  composite  laminated  shell  finite  elements. 
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Orleans,  Louisiana,  pp.  757-758. 

2.  Wang,  B.  and  To,  C.W.S. ,  Finite  element  analysis  of  geometrically 
nonlinear  composite  laminated  plates  and  shells.  Proc.  Second  Int.  Conf.  on 
Composites  Eng.,  August  21-24,  1995,  New  Orleans,  Louisiana,  pp.  791-792. 

3.  To,  C.W.S.  and  Wang,  Hybrid  strain  based  geometrically  nonlinear 
laminated  composite  triangular  shell  elements,  Part  II:  Numerical  studies. 
Comp,  and  Struct.  (Submitted),  1996. 

4.  Liu,  M.L.  and  To,  C.W.S.,  Hybrid  strain  based  three  node  flat 
triangular  shell  elements.  Part  I:  Nonlinear  theory  and  incremental  formulation. 
Comput.  Struct.,  1995,  54,  1031-1056. 

5.  Noor,  A.K.  and  Mathers,  M.D.,  Nonlinear  finite  element  analysis 
of  laminated  composite  shells.  In  Computational  Methods  in  Nonlinear 
Mechanics  (Ed.  by  J.T.  Oden,  E.B.  Becker,  R.R.  Craig,  R.S.  Dunham,  C.P. 
Johnson  and  W.L.  Oberkampf).  Proc.  Int.  Conf  on  Comput.  Methods  in 
Nonlinear  Mechanics,  Austin,  TX,  1974. 

6.  Lin,  J.J.,  Fafard,  M.,  Beaulieu,  D.  and  Massicotte,  B.,  Nonlinear 
analysis  of  composite  bridges  by  the  finite  element  method.  Comput.  Struct., 
1991,40.  1151-1167. 


1453 


7.  Madenci,  E.  and  Bamt,  A.,  A  Free-formulation-based  flat  shell 
element  for  nonlinear  analysis  of  thin  composite  structures.  Int.  J.  Numer. 
Meth.  Engng.,  1994,  37,  3825-3842. 

8.  Bergan,  P.G.,  and  Nygard,  M.K.,  Nonlinear  shell  analysis  using  free 
formulation  finite  elements.  In  Finite  Element  Methods  for  Nonlinear 
Problems,  (edited  by  Bergan,  P.G.,  Bathe,  K.J.,  and  Wunderlich,  W.)  Springer- 
Verlag,  1986. 

9.  Zhu,  J.,  Application  of  natural  approach  to  nonlinear  analysis  of 
sandwich  and  composite  plates  and  shells.  Comput.  Meth.  Appl  Mech.  Engng., 
1995,  120,  355-388. 

10.  Saleeb,  A.F.,  Chang,  T.Y.,  Graf,  W.  and  Yingyeunyong,  S.,  A 
hybrid/mixed  model  for  nonlinear  shell  analysis  and  its  applications  to  large- 
rotation  problems,  Int.  J.  Num..  Meth.  Engng.,  1990  29,  407-446. 

11.  To,  C.W.S.  and  Wang,  B.,  Hybrid  strain-based  three-node  flat 
triangular  laminated  composite  shell  elements  for  vibration  analysis.  J.  Sound 
and  Vibration  (submitted),  1996. 

12.  To,  C.W.S.  and  Wang,  B.,  Transient  response  analysis  of 
geometrically  nonlinear  laminated  composite  shell  structures.  Proc.  of  Design 
Eng.  Conf.  and  Computers  in  Eng.  Conf.  (edited  by  McCarthy,  J.M.),  August 
18-22,  1996,  Irvine,  California,  96-DETC/CIE-1623. 

13.  Reddy,  J.N.,  Geometrically  nonlinear  transient  analysis  of 
laminated  composite  plates.  A.I.A.A.  J.,  1987,  21,  621-629. 

14.  Wu,  C.Y.,  Yang,  T.Y.  and  Saigal,  S.,  Free  and  forced  nonlinear 
dynamics  of  composite  shell  structures.  J.  Comp.  Mat,  1987,  21,  898-909. 

15.  To,  C.W.S.  and  Wang,  B.,  Hybrid  strain  based  geometrically 
nonlinear  laminated  composite  triangular  shell  elements,  Part  I:  Theory  and 
element  matrices.  Computers  and  Structures  (submitted),  1996. 


A  mesh 


D  mesh 


Figure  1  Flat  triangular  laminated  composite  shell  element 


1454 


Central  deflection  lY 


Figure  2  Response  of  a  cross-ply  plate 


Figure  3  Spherical  shell  segment  under  a  uniformly  distributed  load 


1455 


Central  deflection  -W 


Time  b 


Figure  4  Apex  response  with  quarter  shell  considered 


-0,2  0.0  0.2  0.4  0.6  0.0  1.0  1.2  1.4 

X  (m) 

Figure  5  Evolution  of  cantilever  panel  with  free  end  step  moment 


1456 


ANALYTICAL  METHODS  III 


THE  FREE,  IN-PLANE  VIBRATION  OF  CIRCULAR  RINGS 
WITH  SMALL  THICKNESS  VARIATIONS 

R  S  Hwang,  C  H  J  Fox  and  S  McWilliam 
Department  of  Mechanical  Engineering ,  University  of  Nottingham, 
University  Park,  Nottingham  NG7  2RD,  England 

Abstract 

Geometric  imperfections  which  cause  thickness  variations  will  always 
exist  in  nominally  circular  rings  and  cylinders  due  to  limitations  in 
manufacturing  processes.  The  effects  of  circumferential  thickness  variations  on 
the  natural  frequencies  of  in-plane  vibration  are  studied.  The  circumferential 
variations  in  the  inner  and  outer  surfaces  are  describedj  in  a  very  general  wayj 
by  means  of  Fourier  series.  Novozhilov  thin-shell  theory  is  used  in  conjunction 
with  the  Rayleigh-Ritz  method  to  obtain  the  natural  frequencies.  Results  are 
presented  which  show  the  effects  of  single-harmonic  variations  in  the  inner  and 
outer  surface  profiles,  taking  account  of  the  profile  amplitude  of,  and  the  spatial 
phasing  between,  the  inner  and  outer  profiles.  The  frequency  factors  calculated 
from  the  numerical  method  are  in  good  agreement  with  those  obtained  from  the 
Finite  Element  method. 


1.  Introduction 

The  free  vibrations  of  circular  rings  or  shells  had  been  studied  by  many 
authors  for  over  a  century.  The  early  theoretical  works  are  summarised  by 
Love  [I].  Most  of  these  works  are  restricted  to  perfect  rings  or  shells. 
However,  in  practice,  geometric  imperfections  (thickness  variations  and 
departure  from  true  circularity)  are  produced  in  the  manufacturing  process. 
These  affect  the  natural  frequencies  and  mode  shapes.  It  is  weE  known  that  in  any 
truly  axisymmetric  structure  the  vibration  modes  occur  in  degenerate  pairs  which 
have  equal  natural  frequencies  and  mode  shapes  which  are  spatially  orthogonal  but 
of  indeterminate  circumferential  location.  The  main  effects  of  thickness  variations 
are  to  split  the  previous  equal  natural  frequencies  and  remove  the  positional 
indeterminacy  [2].  Although  these  effects  are  often  practically  unimportant,  there 
are  some  applications  (especially  inertial  sensors  based  on  vibration  rings  or 
cylinders  [3] )  where  the  small  frequency  splits  and  fixing  of  the  modal  positions  is 
of  primary  practical  significance.  There  is  therefore  a  requirement  to  be  able  to 
predict  in  detail  the  effects  on  vibrational  behaviour  due  to  small  departures  from 
perfect  circularity  of  the  kind  produced  by  manufacturing  tolerances. 


1457 


The  vibration  of  imperfect  bells  and  rings  were  studied  m  the  general 
way  using  group  theory  [4,5].  In  reference  [5]  the  selection  rules  for  frequency 
splitting  of  thin  circular  rings  were  presented  qualitatively.  In  reference[6]  the 
frequency  splitting  behaviour  of  a  thin  circular  ring  was  investigated  both 
experimentally  and  analytically  by  first  order  perturbation  theory.  In  reference 
[7],  the  classical  frequency  equations,  which  are  generally  used  to  predict  the 
natural  frequencies  of  a  thin  circular  ring,  were  modified  to  describe  an 
eccentric  ring  by  using  the  perturbation  method.  In  reference  [8],  Fourier  series 
functions  were  used  to  represent  the  circumferential  thickness  variations  of  an 
eccentric  cylinder.  Love  thin-sheU  theory,  which  is  only  strictly  suitable  for  a 
perfect  ring  or  cylinder,  was  applied  to  investigate  the  free  vibration  of  non- 
circular  shells. 

In  this  paper,  the  free  in-plane  vibrations  of  thin  rings  of  rectangular 
cross  section  with  circumferential  variations  in  thickness  are  studied.  The 
circumferential  variations  in  the  inner  and  outer  surfaces  are  described,  in  a  very 
general  way,  by  means  of  Fourier  series.  Novozhilov  thin-shell  theory  [9],  in 
conjunction  with  the  well-known  Rayleigh-Ritz  method,  are  applied  to  analyse 
the  vibration  characteristics  for  in-plane  flexural  vibration  of  the  ring  which  is 
considered  as  a  special  case  of  a  thin  shell  [2,6].  The  numerical  method  is  used 
to  investigate  the  effect  of  single-harmonic  circumferential  variations  in  the 
inner  and  outer  surface  profiles.  The  effects  of  harmonic  number,  amplitude  and 
spatial  phasing  between  the  inner  and  outer  profiles  are  investigated.  Some 
important  trends  and  patterns  of  effects  of  profile  variations  on  the  splitting  of 
the  natural  frequencies  are  observed.  The  results  obtained  by  using  the 
numerical  method  developed  in  the  current  investigation  are  validated  by 
comparison  with  Finite  Element  predictions. 


2.  Method  of  Analysis 
2.1  Geometry 

Consider  a  thin  ring  of  mean  radius  having  a  rectangular  cross- 
section  of  mean  thickness  h  («  r«)  and  axial  length  L  («  Ta).  The  inner  and 
outer  surface  vaiy  along  the  global  circumferential  direction  (Figure  1).  rp 
denotes  the  distance  from  the  centre  of  the  mean  radius  of  the  ring  to  the  point 
F  on  the  middle  surface. 

Two  coordinate  systems  are  used  in  the  formulation  of  the  equation  of 
deformation. 


1458 


*  Global  polar  coordinates  (a',  p',  These  are  dir^ted  along  the  global 
axial,  circumferential  and  radial  directions.  The  initial  geometry  of  the 
undeformed,  imperfect  ring  is  defined  using  this  coordinate  system. 

*  Local  curvilinear  coordinates  (a,  p,  These  are  directed  along  the  local 
axial,  tangential  and  normal  directions  relative  to  the  true  middle  surface  and 
coincident  with  the  principal  coordinates  of  the  middle  surface.  This  local 
coordinate  system  is  required  for  implementation  of  Novozhilov  shell  theory 
which  specifies  displacements  in  the  local  tangential  and  normal  directions. 

is  the  angle  between  the  global  and  local  coordinate  systems  at  the  point  P 
of  the  middle  surface. 

All  the  displacements,  thicknesses,  and  radii  in  this  paper  are  expressed 
dimensionlessly  by  dividing  by  lo ,  where  k  is  the  representative  length  and  is 
defmed  as  the  mean  radius  of  the  ring. 


Figure  1.  A  thin  ring  having  circumferentially  arbitrary  surfaces 

The  shape  of  the  middle  surface  of  the  ring  is  determined  by  the  inner 
and  outer  surfaces  which  can  be  expressed  by  Fourier  series  as  follows: 

f*(?')=U+'^f*cos(ip)  +  '^f;  sin(jp)  (1) 

i=I  j=I 

/TP’)=/«'+ X/r«osc«P'-)+X/7««o‘P')  (2) 

i=/ 


1459 


where  /  and  /  "(P')  denote  respectively  the  outer  and  inner  surface 
functions  with  respect  to  the  global  circumferential  coordinate  P  and  fo  ,//  j 
//,  fo',  fi'  and//  are  the  Fourier  coefficients  which  are  defined  in  the  usual 
way  [10]. 

The  middle  surface  of  a  shell  or  ring  is  defined  as  the  locus  of  the  points 
which  lie  at  equal  distances,  *  and  from  the  outer  and  inner  surfaces  along 
the  direction  normal  to  the  mid  surface  (see  Figure  2). 


..  ^ 


Figure  2.  The  bounding  surfaces  and  the  middle  surface 


For  given  inner  and  outer  surfaces,  /  (PpO  and/'^(p/»0  defined  in  the 
global  coordinate  system,  the  true  middle  surface  can  be  determined  using  an 
iterative  numerical  procedure  which  is  fully  described  in  [1 1].  Once  the  point  P 
on  the  true  middle  surface  has  been  determined,  the  corresponding  ,  Rp, 

Tp  and  Pp  can  be  calculated-  These  will  be  used  in  the  step-by  step  integrations 
which  determine  the  strain  energy  and  kinetic  energy  of  the  ring. 


2.2  Equations  of  Motion 


The  strain  energy  for  a  thin  ring  whose  length  is  much  smaller  than  the 
mean  radius  takes  the  form  [2,6]: 

S  =  «(!->■  l/R)d^d^  (3) 


1460 


Based  on  Novozhilov  thin-shell  theory,  the  normal  strain  epp  in  equation 
(3)  is  given  as 


"  7  t  /  T>  ^ 

(4) 

1  +  ^/  R 

Ep=  Vj^/R  +  w/R 

(5) 

Kp=  -  1/R  ( IV,  p/j?  +v/R),  p 

(6) 

where  Ep,  Kp  characterise  the  deformation  of  the  middle  surface  of  the  thin  ring 
and  subscript  “  ,  p  “  denotes  partial  derivatives  with  respect  to  (5.  Epis  the  strain 
tangential  to  the  middle  surface  and  Kp  is  the  change  of  curvature,  v,  w  are  the 
nondimensional  local  displacement  components  of  the  point  P  on  the  middle 
surface  along  the  tangential  and  normal  directions  respectively. 

Substituting  equations  (4)-(6)  into  equation  (3),  then  integrating  with 
respect  to  the  thickness  from  hi~  to  neglecting  the  4th  and  higher  powers 

of  /r F ^  and  ^  f  and  noting  that  F(p)  [ -  hi~^] d?t  =  0  where  Ffp)  is 

an  arbitrary  function  of  p,  the  strain  energy  of  a  thin  ring  can  be  derived  in 
terms  of  the  local  displacements  v  as  follows: 

S=  {[(  V,  p/  +  2h'v,  p  +  w^](  1  /R)[hi*  -hi~  ] 

2  rp; 

jK 

+  [ 2WW,  p  +  2w,  p  IV,  pp  -  2v  w  -  2viv,  pp  ]  -h 

3K 

+  [(w,^)^  +  v‘.2vw,f]^  ((l/R),f)^[kC^  -hr^]}d^  (7) 

jK 

Similarly,  the  kinetic  energy  of  a  thin  ring,  based  on  Novozhilov  thin- 
shell  shell  theory,  can  be  expressed  as  follows: 

T=  {[(y,,)\(y^,,f]R[hC  + 

2  '•Pi 

[3(v,t  f-  4v,tW,,p  +  (  IV, /p  f  f  (8) 

in  which  p  is  the  density  of  the  ring,  and  the  subscript  ‘ ,/  *  denotes  the  partial 
derivatives  with  respect  to  time. 


1461 


For  free  vibration  the  tangential  displacement  v  and  the  normal 
displacement  w  'which  satisfy  the  boundary  condition  can  be  assumed  to  take  the 
following  forms  reg)ectively: 


V  =  sinnp-v^  cosnp)^^^^  (9) 

n 

w  =  cosnp  +  w^  sinnp)^^^^  (10) 

n 

where  v„  and  are  the  undetermined  amplitude  coefficients  of  the  tangential 
and  normal  displacements  of  the  middle  surface  respectively.  The  superscripts 
"s"  and  "c”  refer  to  the  fact  that  these  coefficients  are  multiplied  by  sine  and 
cosine  terms  respectively. 

Substituting  equations  (9)  and  (10)  into  equations  (7)  and  (8),  then 
applying  the  Rayleigh-Ritz  procedure,  the  general  frequency  equation  of  the 
free  ■vibrations  for  a  thin  ring  is  obtained  and  can  be  expressed  in  the  following 
general  matrix  form: 


-A 

'M" 

Ui~ 

~0~ 

0 

where  jfi:  and  M  represent  stiffiiess  and  mass  matrices  of  size  2(N+1),  and  q 
denotes  a  vector  of  generalised  coordinates  v„,  w„  etc.  The  matrix  elements  in 
equation  (11)  are  given  in  [11].  Since  in  the  general  case  the  cross-section  of 
the  ring  'wdl  not  be  symmetric  with  respect  to  P  =  0,  the  classification  of  the 
modes  as  being  “symmetric”  and  “antisymmetric”  is  meaningless.  In  the  special 
case  of  a  perfect  circular  ring,  the  off-diagonal  terms  [K^J,  /M"/  and 
/Af7  appearing  in  equation  (11)  are  null  matrices,  then  equation  (11)  can  be 
uncoupled  into  two  equations:  one  is  for  the  symmetric  modes  and  the  other  is 
for  the  anti- symmetric  modes  -with  respect  to  P=0. 


The  frequency  factors  of  the  ring.  An  ,  are  the  eigenvalues  of  equation 
(11)  and  are  defined  by 


A 


0)1  d  P 
E 


(12) 


where  a)„  is  the  natural  frequency  of  the  nth  radial  mode.  The  frequency  factor 
An  is  proportional  to  the  square  of  natural  frequency  . 


For  a  given  value  of  n  equation  (11)  will  yield  a  pair  of  values  of  An. 
These  will  be  equal  in  the  case  of  a  perfect  ring  but  will  be  slightly  different  in 


1462 


the  case  of  an  imperfect  ring,  giving  rise  to  a  higher  frequency  mode  and  a 
lower  frequency  mode  for  each  value  of 

It  should  be  noted  that  the  matrix  elements  in  equation  (11)  are 
expressed  as  integrals  of  the  functions  ,  Rp,  yp  and  ^p  with  respect  to 

the  tangential  coordinate  These  functions  are  expressed  in  terms  of  the  local 
coordinates.  Hence  it  is  necessary  to  make  a  transformation  to  express  these 
functions  and  integrals  in  terms  of  the  global  coordinates,  so  that  the  integrals 
can  be  evaluated  over  the  global  circumferential  coordinate  p*  from  0  to  27t. 


3.  Results  and  Discussion 

By  using  different  combinations  of  trigonometric  functions  in  equations 
(1)  and  (2),  it  is  in  principle  possible  to  model  any  closed  thin  ring.  For  the 
purpose  of  illustration  we  will  consider  a  nominally  circular  ring  with  a  single 
harmonic  variation  in  the  inner  and  outer  surfaces,  given  by 

(P')  ~ 

/"  OT  =  +  V  (JP’  -  <!>) 

where  h/  and  hf  are  the  ampHtudes  of  the  unperfections  of  the  outer  surface 
and  the  inner  surface  measured  from  the  mean  outer  radius  ra  and  the  mean 
inner  radius  r/  respectively,  is  the  spatial  phase  angle  between  the 
trigonometric  functions  of  the  inner  and  outer  surfaces  at  p*  =0  ,  and  /,  j  are  the 
harmonic  numbers  of  the  surface  variations.  Figure  3  illustrates  i  =]  =  3  for 
three  values  of  ^  . 


(^  =  0 


<j>  =  7t/2 


^  % 


Figure  3.  Different  spatial  phase  angles  ^  for  i  =  j  =  3 


1463 


Results  for  the  combinations  of  the  geometric  imperfections  of  i  -J  -  2, 
3,  4,  5,  6,  hf*  -  hf  =  O.lhy  OMlh  ,  and  <J»  =  0, 7c/4,  nil,  SjtM,  tz  are  presented 
here.  The  ring  dimensions  and  material  properties  are  as  follows:  = 

40.75mm,  r/  =  37.83mm,  L  =  2  ram  =  206.7x10’  NW,  p  =  7850  kg/ml 

Note  that  hf  =  O.lh  corresponds  to  a  departure  from  circular  which  is 
much  larger  than  would  occur  in  practice  due  to  imperfection.  The  results  for  hf  — 
OJh  are  presented  to  highlight  the  effects.  Practically  however,  hf  =  OJDlh 
represents  a  more  realistic  variation  in  thickness. 

Convergence  studies  indicated  that  for  hf* ,  hf~  =  O.lh  the  use  of  30  terms 
in  the  solution  series  (equations  9  and  10)  gave  4  significant  figure  accuracy  or 
better  for  the  frequency  factors  A*  fox  k  -  0, 1,  ...  6.  This  was  considered  to  be 
acceptable  for  the  purposes  of  the  illustrative  examples  considered  here. 

In  a  parallel  Finite  Element  study,  beam  elements,  and  two-  and  three- 
dimensional  plane  stress  elements  used  to  model  an  imperfect  ring.  In  order  to 
get  4  significant  figures  or  better,  120  elements  were  used  to  model  the 
complete  ring.  Comparison  of  the  results  obtained  from  the  numerical  method 
and  the  Finite  Element  Method  shows  that 

(i)  there  is  good  agreement  between  the  curves  of  frequency  factors 

obtained  by  the  Finite  Element  method  and  the  numerical  method. 

(ii)  the  trends  and  patterns  of  frequency  splitting  are  nearly  identical 

irrespective  of  the  analysis  methods  or  the  types  of  finite  elements  used. 

In  considering  the  effect  of  single  harmonic  variations  of  the  profile  of  the 
inner  and  outer  surfaces  on  the  natural  fiequencies  of  different  radial  modes,  the 
discussion  will  focus  on  three  aspects: 

(a) ,  the  effect  of  the  harmonic  number  of  the  profile; 

(b) .  the  effect  of  the  magnitude  of  the  profile  variations;  and 

(c) .  the  effect  of  the  spatial  phasing  between  the  profile  variations  of  the 

inner  and  outer  surfaces. 

The  frequency  splits  shown  are  often  very  small  (~  0.001%).  Note 
however  that  in  some  inertial  sensor  applications,  such  small  frequency  splits 
may  be  of  practical  significance. 


1464 


(a)  The  effect  of  profile  harmonic  number 

Table  1  compares  the  frequency  factors  A„  obtained  for  a  perfect  ring 
and  an  imperfect  ring  for  ^-0,  "^12,  tc,  h/"  -hj  —  0,lh,  and  i=j  =  2  to  6.  It  is 
evident  from  these  results  that: 

For  the  flexural  modes  {n>2): 

(i)  When  i  ,  j  are  equal  and  even  (see  Table  1),  frequency  splitting  only 
occurs  in  the  nth  mode  where  w  =  Id.  I  2  and  /c  is  an  integer.  The 
maximum  frequency  splitting  occurs  in  the  n—H  2  modes  (i.e.  k  ^  1  ) 
and  the  splitting  decays  as  k  increases. 

(ii)  When  i  ,  j  are  equal  and  odd  (see  Table  1)  frequency  splitting  only 
occurs  in  the  nth  mode  where  n  =  ki  and  k  ism  integer.  The  maximum 
frequency  splitting  occurs  for  A:  =  1,  and  splitting  decreases  as  k 
increases. 

It  should  be  noted  here  that  frequency  splitting  in  the  higher  modes 
exists  but  is  very  small,  e.g.,  for  <}»  =  0,  i  -  j  =  2  and  kf=  OJh  ,  frequency 
splitting  occurs  in  the  2nd  and  higher  radial  modes.  It  can  be  seen  from  Table  1 
that  the  splits  in  frequency  factor  are  0.019%  at  the  2nd  mode,  0.001%  at  the 
3rd  mode,  and  less  than  0.001%  at  the  4th  mode  or  higher  mode.  These 
correspond  to  actual  frequency  splits  of  about  0.01%,  0.(XK)5%  and  less  than 
0.CKX)5%  respectively  (equation  (12)). 

For  the  radial  extensional  mode( «  =  0 ),  no  frequency  splitting  occurs. 

It  is  clear  from  Table  1  that  the  trends  and  patterns  of  frequency  splitting 
are  the  same  for  ^-0,n/2  and  tc.  However,  frequency  splitting  is  less  foT^  =  0 
than  for  <})  =  71  under  the  same  conditions.  Frequency  splits  for  ^  between  0  and 
K  are  intermediate  between  those  for  <{»  =0  and  <{)  =  7t . 

The  above  patterns  are  in  agreement  with  the  qualitative  results 
published  in  reference  [5]  in  which  only  the  conditions  for  non-splitting  are 
established. 


1465 


Table  1 


The  difference  of  frequency  factors  A  on  the  radial 
modes  w(n) [the  parameters  of  profile  variations  are 
taken  as  hf=0.1h/  i=j=2  to  $  and  (a)«|>=0;  (b)i^5=TC/2;  (c)4>=?c] 


(a)  4>= 

=0 

A(0) 

A(2) 

A{3) 

A(4) 

A(5) 

A(6) 

Perfect 

1.0005 

0.003313 

0.02645 

0.09693 

0.2525 

0.5406 

i=  2 

high 

low 

-0.011% 

-0.023% 

-0.042% 

-0.041% 

-0.042% 

-0,038% 

-0.036% 

-0.035% 

i=  3 

high 

low 

0.075% 

-0.047% 

-0.088% 

-0.153% 

-0.133% 

-0.108% 

-0.094% 

-0.095% 

i=  4 

high 

low 

0.457% 

2.155% 

-2,287% 

-0.134% 

-0.150% 

-0.594% 

-0.349% 

-0.240% 

-0.248% 

i=  5 

high 

low 

1.631% 

-0.171% 

-0.141% 

-0.332% 

-0.229% 

-1.816% 

-0.838% 

i=  6 

high 

low 

5.656% 

-0.171% 

2.406% 

-2.726% 

-0.267% 

-0.715% 

-0.302% 

-5’.624% 

(b)  4)=Tr/2 

A(0) 

A(2) 

A{3) 

A(4) 

A{5) 

A(6) 

i= 

2 

high 

low 

0.007% 

0.336% 

-2.427% 

-0.595% 

-0.676% 

-0.971% 

-0.972% 

-1.169% 

-1.272% 

i= 

3 

high 

low 

0.090% 

-1.231% 

0.336% 

-2.559% 

-0.721% 

-0.869% 

-1.038% 

-1.040% 

i= 

4 

high 

low 

0.382% 

14.58% 

-18.71% 

-1.150% 

0.287% 

-2.871% 

-0.868% 

-0.828% 

-0.901% 

i  = 

5 

high 

low 

1.236% 

-5.642% 

-1.239% 

-1.201% 

0.212% 

-3.703% 

-1.172% 

i  = 

6 

high 

low 

4.111% 

-5.620% 

13.12% 

-17.59% 

-1.316% 

-1.404% 

0.117% 

-6.370% 

(c)  41=71 


i  = 

2 

high 

low 

A{0) 

0.026% 

A(2) 

0.690% 

-4.827% 

A(3) 

-1.132% 

-1.340% 

A(4) 

-1,908% 

-1.912% 

A(5) 

-2.308% 

A{6) 

-2.514% 

i  = 

3 

high 

low 

0.106% 

-2.469% 

0.709% 

-4.934% 

-1.312% 

-1.637% 

-1.989% 

-1.993% 

i= 

4 

high 

low 

0.314% 

18.81% 

-26.96% 

-2.227% 

0.528% 

-4.993% 

-1.388% 

-1.447% 

-1.532% 

i= 

5 

high 

low 

0.881% 

-10.89% 

-2.400% 

-2.135% 

-0,018% 

-5.031% 

-1.510% 

i= 

6 

high 

low 

2.808% 

-10.81% 

16.78% 

-25.36% 

-2.396% 

-2.134% 

-1.881% 

-5.060% 

Note: 


1.  difference  =  [A  (n)  -  A  (n)porfc.ct]x  100%  /  A  (n) perfect 

i\o 

2.  A(n)  =  — — (H  (n)  ,  where  (ii(n)  is  the  natural  frequency  at 

E 

the  nth  radial  mode. 


1466 


Table  2.  The  difference  of  frequency  factors  A  on  the  radial 
inodes  w(n)  [the  parameters  of  profile  variations  are 
taken  as  hf=0.01h,  i=j=2  to  6  and  (a)<|>=0;  (b)<|)=TC/2;  (c)<|)=7t  ] 


(a)  (|)=0 

Perfect 

A(0) 

1.0005 

A(2) 

0.003313 

A(3) 

0.02645 

A{4) 

0.09693 

A(5) 

0.2525 

A(6) 

0.5406 

i=  2  high 
low 

=0 

=0 

=0 

=0 

=0 

=0 

i=  3  high 
low 

0.001% 

«0 

-0.001% 

-0.002% 

-0.001% 

-0.001% 

-0.001% 

i=  4  high 
low 

0.005% 

0.222% 

-0.223% 

-0.001% 

-0.001% 

-0.006% 

-0.004% 

-0.002% 

i=  5  high 
low 

0.016% 

-0  .  002% 

-0.002% 

-0.003% 

-0 . 002% 
-0.019% 

-0.009% 

i=  6  high 
low 

0.060% 

-0.002% 

0.256% 

-0.260% 

-0.003% 

-0.008% 

-0.003% 
-0-.  063% 

(b)  <j)=7t/2 

A{0) 

A(2) 

A(3) 

A(4) 

A(5) 

A(6) 

i=  2 

high 

low 

=0 

0.004% 

-0.024% 

-0.006% 

-0.010% 

-0 . 012% 

-0.013% 

i=  3 

high 

low 

0.001% 

-0.012% 

0.003% 

-0.026% 

-0.007% 

-0.009% 

-0.010% 

i=  4 

high 

low 

0.004% 

1.684% 

-1.725% 

-0.011% 

0.003% 
-0 . 029% 

-0 . 009% 

-0.009% 

i=  5 

high 

low 

0.012% 

-0 . 058% 

-0 . 012% 

-0.012% 

0.002% 

-0.038% 

-0.012% 

i=  6 

high 

low 

0.044% 

-0.057% 

1.550% 

-1.595% 

-0 . 013% 

-0.014% 

0.001% 

-0.069% 

(C)  <j>=7t 


i=  2 

high 

low 

A{0) 

=0 

A(2) 

0 . 008% 
-0  .  048% 

A(3) 

-0.012% 

A(4) 

-0.019% 

A(5) 

-0.023% 

A(6) 

-0.025% 

i=  3 

high 

low 

0.001% 

-0  .  024% 

0.008% 

-0.050% 

-0 . 013% 

-0.016% 

-0.020% 

i=  4 

high 

low 

0.003% 

2.360% 

-2.440% 

-0.021% 

0.006% 
-0 . 050% 

-0.014% 

-0.015% 

i=  5 

high 

low 

0.009% 

-0.114% 

-0.023% 

-0.020% 

«0 

-0.051% 

-0.015% 

i=  6 

high 

low 

0.028% 

-0.112% 

2.165% 

-2.251% 

-0.023% 

-0.020% 

-0 . 018% 
-0 . 051% 

Note:  1.  difference  =  [A  (n)  -  A  (n)p*r£ecc]x  lOO^s  /  A(n}p<,r£«ot 

2.  A(n)  =  (£t^(n) .  where  tOfn)  is  the  natural  frequency  of 

E 

the  nth  radial  mode. 


1467 


(b)  The  elBfect  of  proBle  amplitude 

The  effects  of  varied  profile  amplitude  (h/'  =  hf  =  O.lh  and  OMlh)  upon 
the  frequency  factors  for  ^  =  0,  it  and  i  =J  =  2  to  6  can  be  seen  by 
comparing  Tables  1  and  2.  It  may  be  concluded  from  these  results  and  others 
which  are  presented  in  [1 1]  that: 

(1)  When  n-i  1 2,  frequency  factor  splitting  due  to  variable  profile  magnitude 

compared  with  the  frequency  factor  of  the  perfect  ring  is  nearly 
proportional  to  the  profile  amplitudes,  hj*  and  hj  . 

For  example,  for  i  ^  =  7t  and  hf^  =  hf  0,lh  ,0.01h  (see  Tables  1 

and  2) ,  the  magnitude  of  frequency  splitting  at  the  2nd  mode  is  45.77%  for 
hf  =  OJh  and  4.80%  for  hf=  OMlh.  These  correspond  to  actual  frequency 
splits  of  24%  and  2.4%  respectively  (equation  (12)) 

(2)  For  modes  other  than  those  for  which  n^i  12,  splitting  of  frequency  factors 

is  nearly  proportional  to  the  square  of  the  profile  amplitudes,  and  hf  . 

For  example,  for  i  “4 ,  2nd  hf*  =  hf  —  O.lh  ,0Mlh  (see  Tables  1 
and  2),  the  magnitude  of  frequency  factor  splitting  at  the  4th  mode  are 
5.521%  for  hf- O.lh  and  0.056%  for  A/=  O.Olh. 

These  results  shown  in  Tables  1  and  2  for  (j)  =  0  and  =  ^t/2  show  that 
the  general  nature  of  the  trends  regarding  the  effect  of  profile  amplitude 
variations  on  the  frequency  factors  are  the  same  for  all  values  of  ^  ,  although 
the  magnitudes  of  the  changes  in  frequency  factors  depend  on  ({>,  as  discussed  in 
the  following  section. 

(c)  The  effect  of  spatial  phase  angle  variations 

The  effects  of  the  variations  of  spatial  phase  angle  <j)  on  the  frequency 
factors  are  shown  in  Figure  4,  from  which  it  is  evident  that 

(1)  As  frequency  splitting  occurs  (see  Figure  4.a-4.d  ),  the  maximum  frequency 
splitting  is  obtained  at  <)  =  tc  and  the  minimum  splitting  occurs  at  (p  =  0.  It 
is  clear  that  the  maximum  frequency  splitting  occurs  in  the  n  —  il2  modes. 

(2)  In  modes  for  which  no  frequency  splitting  occurs  (see  Figure  4.e  and  4.f ), 

the  minimum  frequency  difference  compared  with  that  of  the  perfect  ring  is 
detected  at  <j)  =  0  and  the  maximum  at  ^  =  n.  Irrespective  of  the  value  of 
(j)  ,  the  frequencies  of  these  modes  are  always  less  than  the  corresponding 
frequencies  of  the  perfect  ring. 


1468 


WKemnceIn  4  (%)  at(eiwic»ln  A  (%) 


Figure  4.  Effect  of  spatial  phasing  on  frequency  factors  with  hf  =  O.lh  and 
(a)  i=j=:4, 2nd  mode ;  (b)  i=:j=6, 3rd  mode 
(c)  i=j=2,  2nd  mode  ;  (d)  i=j=3, 3rd  mode 
(e)  i=j=3,  4th  mode  ;  (f)  i=j=4,  5th  mode 


1469 


4,  Conclusions 


In  this  paper,  Novozhilov  thin-shell  theory  and  the  Rayleigh-Ritz  procedure 
have  been  applied  to  derive  the  frequency  equations  of  a  thin  ring  with  a  rectangular 
cross-section  and  a  circumferential  proffle  variatioa  Profile  variations  are 
represented,  in  the  general  way,  by  Fourier  series  functions  and  the  method  gives 
quantitative  predictions  of  frequency  splitting.  The  observed  firequency  splitting 
patterns  are  in  agreement  with  previously  published  qualitative  results.  Numerical 
results  have  been  presented  for  example  cases  in  which  the  inner  and  outer  profiles 
are  nominally  circular  with  superimposed  single-harmonic  variations  m  radius.  The 
effects  on  frequency  splitting  of  the  harmonic  number  of  the  profile  variation,  and 
the  amplitude  and  spatial  phasing  between  the  inner  and  outer  surfaces  have  been 
investigated. 


References 

1.  Love,  A-E.H.,  A  Treatise  on  the  Mathematical  Theory  of  Elasticity^  Dover 

Publications,  New  York,  fourth  edition,  1952. 

2.  Fox,  A  simple  theory  for  the  analysis  and  correction  of  frequency 

splitting  in  slightly  imperfection  rings.  Journal  of  Sound  and  Vibration, 
1990, 142(2),  227-43. 

3-  Fox,  Vibrating  cylinder  rate  gyro:  theory  of  operation  and  error 

analysis.  Proceeding  of  DGON  Symposium  on  Gyro  Technology, 
Stuttgart,  1988,  Chapter  5. 

4.  Chamley,  T.  and  Perrin,  R.,  Studies  with  an  eccentric  bell.  Journal  of  Sound 

and  Vibration,  1978,  58(4),  517-25. 

5.  Perrin,  R.,  Selection  rules  for  the  splitting  of  the  degenerate  pairs  of  natural 

frequencies  of  thin  circular  rings.  Acustica,  1971,  25,  69-72. 

6.  Chamley,  T.  and  Perrin,  R.,  Perturbation  studies  with  a  thin  circular  ring. 
Acustica,  1973,  28,  139-46. 

7.  Valkering,  T.P.  and  Chamley,  T.,  Radial  vibrations  of  eccentric  rings. 
Journal  of  Sound  and  Vibration,  1983, 86(3),  369-93. 

8.  Tonin,  R.F.  and  Bies,  D.A.,  Free  vibration  of  circular  cylinders  of  variation 

thickness.  Journal  of  Sound  and  Vibration,  1979,  62(2),  165-80. 

9.  Novozhilov,  V.V.,  The  Theory  of  Thin  Shells,  P.  Noordhoff  Ltd.,  The 
Netherlands,  1959. 

10. Kreyszig,  E.,  Advanced  Engineering  Mathematics,  John  Wiley  &  Sons, 
Inc.,  Singapore,  1993,  pp.  569-71. 

11. Hwang,  R.,  Free  vibrations  of  a  thin  ring  having  circumferential  profile 
variations.  Ph.D.  Thesis,  University  of  Nottingham,  U,K,  (in  preparation) 


1470 


FREE  VIBRATION  ANALYSIS  OF  TRANSVERSE-SHEAR 
DEFORMABLE  RECTANGULAR  PLATES 
RESTING  ON  UNIFORM  LATERAL  ELASTIC  EDGE  SUPPORT 

D.J.  Gorman 

University  of  Ottawa 
770  King  Edward  Ave., 

Ottawa,  Canada  KIN  6N5 


ABSTRACT 

Utilizing  the  Superposition  Method  a  free  vibration  analysis  is  conducted  for 
transverse-shear  deformable  rectangular  plates  resting  on  uniformly  distributed 
lateral  elastic  edge  support.  Edges  are  free  of  moment.  The  thick  isotropic 
Mindlin  plate  is  utilized  for  illustrative  purposes.  The  Mindlin  equations  are 
satisfied  throughout.  Typical  computed  results  are  plotted  for  a  square  plate. 


INTRODUCTION 

It  is  well  accepted  that  classical  rectangular  plate  boundary  conditions  denoted 
as  simply  supported  or  clamped  are  often  not  achieved  in  real  structures.  This  is 
because  of  elasticity  in  the  edge  supports.  Furthermore,  in  many  rectangular 
plate  installations  elastic  edge  supports  may  be  utilized  intentionally.  For  this 
reason  a  number  of  studies  of  effects  of  elasticity  in  the  edge  supports  on 
rectangular  plate  free  vibration  frequencies  have  been  conducted  and  results 
published.  Almost  all  of  these  studies  have  been  devoted  to  the  free  vibration 
behaviour  of  thin  isotropic  plates.  Studies  by  the  author,  related  to  this  family  of 
vibration  problems,  have  been  devoted  to  situations  where  elastic  stiffness  is 
uniformly  distributed  along  the  edges  as  well  as  cases  where  the  stiffnesses  are 
arbitrarily  distributed.  All  of  his  studies  have  been  conducted  by  means  or  the 
Superposition  Method  and  in  a  fairly  recent  article  he  has  demonstrated  that  all 
of  these  families  are  amenable  to  analytical  type  solutions  [1]. 

In  this  paper  we  exploit  the  powerful  Superposition  Method  to  analyse  the  free 
vibration  behaviour  of  transverse-shear  deformable  plates  resting  on  uniform 
lateral  elastic  edge  support.  This  represents  a  much  more  complicated  problem 
than  the  thin  isotropic  plate  problems  discussed  above.  For  our  purposes  we 
choose  the  thick  shear-deformable  Mindlin  plate  and  base  our  solution  on 
Mindlin  theory. 

In  the  interest  of  keeping  the  literature  review  up  to  date  the  recent  publication 


1471 


of  SAHA,  KAR,  and  DATTA  [2]  is  drawn  to  the  attention  of  the  reader.  They 
report  on  a  study  of  thick  Mindlin  plates  resting  on  edge  supports  with  uniform 
lateral  and  rotational  elasticity.  They  have  employed  a  Rayleigh-Rite  energy 
approach.  Plate  lateral  displacement  is  represented  by  a  rather  complicated  set 
of  Timoshenko  beam  functions,  each  extremity  of  each  beam  being  attached  to 
a  local  lateral  and  torsional  spring.  It  will  be  seen  that  no  such  functions  need  be 
selected  in  the  superposition  approach  adopted  here.  Another  related  paper  is 
one  by  the  present  author  dealing  with  Mindlin  plates  where  lateral  displacement 
along  the  plate  edges  is  forbidden  but  uniform  rotational  elastic  support  is 
provided  [3].  This  problem  is  somewhat  easier  to  solve  since  edge  lateral 
displacement  is  forbidden  and,  unlike  the  present  problem,  mixed  derivatives  do 
not  show  up  in  the  boundary  condition  formulation.  This  latter  problem  was 
shown  to  be  amenable  to  solution  by  the  modified  Superposition-Galerkin 
Method  which  is  extremely  easy  to  use  when  it  is  applicable. 


MATHEMATICAL  PROCEEDURE 

A  solution  to  the  present  problem  is  obtained  through  th&  supei-position  of  the 
eic^ht  edge-driven  forced  vibration  solutions  (building  blocks)  shown 
scheraaticaly  in  Figure  1.  All  of  the  non-driven  edges  have  slip-shear  support. 
This  type  of  support,  indicated  in  the  figure  by  two  small  circles  adjacent  to  the 
edc^e,  implies  that  the  edge  is  free  of  torsional  moment  and  transverse  shear 
fomes.  Furthermore,  rotation  of  the  plate  cross-section  along  the  edge  is 
everywhere  zero. 

We  begin  by  examining  the  first  building  block.  Its  driven  edge  is  free  of 
torsional  moment,  and  rotation  of  the  plate  cross-section  along  this  boundary  is 
every  where  zero.  This  latter  condition  is  indicated  by  two  solid  dots  adjacent  to 
the  edge.  Driving  of  this  edge  is  accomplished  by  a  distributed  harmonic 
transverse  sheai*  force  of  circular  frequency  o).  The  spatial  distribution  of  the 
shear  force  is  expressed  as, 

Q,l,.,=  E  E„cos(m-l)7t5  (1) 

m  =  1 ,2 


where  k  is  the  number  of  terms  required  in  the  series. 


1472 


Fig.  1  Schematic  representation  of  building  blocks  utilised  in  theoretical 

analysis. 

We  now  examine  the  response  of  the  above  building  block  to  this  harmonic 
excitation.  The  proceedure  followed  is  almost  identical  to  that  described  in  an 
earlier  publication  [4].  A  concise  description  will  be  provided  here  for  the  sake 
of  completeness. 

The  governing  differential  equations  which  control  the  response  of  thick  Mindlin 
plates  are  written  in  dimensionless  form  as, 

W  d"  W  ^  ^  ^  W  =  0  (2) 

8^“  (j)-  art“  ({)  ari  V3 


d^-  4)-  8  Ti^  4)  5  ^  8  r|  i  ^  5  U 


(3) 


8f  4)^v,  a-n-  4)Vi  d^dr\ 


1473 


Transverse  shear  forces,  bending  moments,  etc.,  are  written  as. 


V  9  „ 


^  3^  cj)  3r|  ^  3ti 


(j)  3n 

at  a  tf  9  <|i-  1  a  i(r. 


When  subjected  to  the  first  term  of  the  driving  force  (Eqn  1)  the  response 
of  the  building  block  will  be  essentially  that  of  a  Timoshenko  beam.  The 
governing  differential  equations  reduce  to  a  set  of  two  which  may  be 
written  as 


d“W  ^  d-ijj  A'^(})“c|);;W 

- +  (p - !•  + - 

d  rt-  d  Ti"“  V3 


=  0 


(6) 


and 

d'l};^  V3<{)V  ^  1  dW  K 

^2  dn  j  12 


(7) 


It  is  convenient  to  represent  the  lateral  displacement  W,  and  plate  cross- 
section  rotation  as, 

W(Ti)  =  X(Ti),  and  (ti)  =  Z  (t)) 

The  governing  differential  equations  may  then  be  written  as 

X"(q)  +  a^,Z'(q)  +  b„,X(ti)  =  0  (8) 


and 

Z''(q)  +  a,^3X'(n)  +  b„,5Z(Ti)  =  0  (9) 

where  superscripts  imply  differentiation  with  respect  to  r|.  Coefficients 


1474 


....  etc,,  are  defined  in  reference  [4]. 

Applying  the  appropriate  differential  operators  to  this  set  of  equations 
the  parameter  X  (ti)  is  eliminated  and  a  second  order  ordinary 
homogenous  differential  equation  is  obtained  involving  the  parameter  Z 
(q).  It  is  found  that  for  our  range  of  interest  the  roots  of  the  characteristic 
equation  associated  with  this  differential  equation  are  always  real.  There 
are  then  three  possible  pairs  of  roots  depending  on  the  coefficients  in  the 
above  differential  equation.  Designating  these  pairs  of  real  roots  as 
Rj  and  possible  forms  of  solution  exist  as  follows, 

Casel,  RpR2<0-0  Case2,  R,<0-0;  R,>0-0  CaseS,  Ri,R2>0-0  (lO) 

In  all  work  reported  here  it  has  been  found  that  only,  case  2,  has  been 
encountered. 

It  will  be  obvious  that  the  functions  X  (q)  and  Z  (q)  must  be  symmetric 
with  respect  to  the  ^  axis.  We  may  therefore  write  for  case  2, 

X(q)  =  cos  aq  +  cosh  pq,  (11) 


and 

Z(^)  =  Am  (  cos  a  q  +  cosh  P  q  (12) 

where  a  =  ^|R, |  ,  and  P  = 

Expressions  and  S^.^re  obtained  by  taking  advantage  of  the 
coupling  of  equations  8  and  9,  as  was  done  in  Reference  [4]. 

We  then  impose  the  boundary  conditions,  Q^  =  E,^^,and  ilf^  =  0,at  q=l, 
in  order  to  evaluate  the  unknowns  and  B^^^  of  Equation  (11). 
Accordingly  we  obtain, 

E 

X(q)  =  .^{cos  aq  +  X  1  cosh  Pq},  (13) 


and 


1475 


Z(T1)  =  ^  {S„ ,  sin  a  n  +  XI  S„,  sinh  P  t|},  (14) 


where  XI  and  X2  are  easily  evaluated. 

Next  we  examine  the  response  of  the  first  building  block  to  driving  terms 
where  m>l.  We  follow  the  proceedure  described  in  Reference  [4]. 


Levy  type  solutions  for  the  parameters  W,  .... 

.  etc.,  are  written  as. 

W(5,ti)  =  XjTi)  cosirni? 

(15) 

litres, Tl)  =  Y„(ri)  sinm-ii^ 

•  (16) 

=  Z„(T|)  cosmn? 

(17) 

It  will  be  noted  that  all  required  boundary  conditions  along  the  edges,  ^ 
=  0,  and  ^  =  1/  are  satisfied. 


Next,  the  above  expressions  (Eqns  15, 16, 17)  are  substituted  in  the  set  of 
governing  differential  equations.  The  following  set  of  coupled  ordinary 
differential  equations,  written  in  matrix  form  are  obtained 


X 

0 

0 

X./' 

t>n,l 

^m2 

0 

x„ 

o' 

Y 

^  m 

0 

0 

^m2 

Y.' 

^m4 

0 

Y„, 

► _  4 

0 

zj 

.  m  J 

.^m3 

^m4 

0 

0 

0 

bn,5. 

0 

(18) 


Again,  the  quantities  ....  etc.,  are  defined  in  Reference  [4]. 

Applying  the  appropriate  operators  on  the  above  equations,  as  was  done 
in  Reference  [4]  we  are  able  isolate  a  single  homogenous  sixth  order 
ordinary  differential  equation  involving  the  dependent  variable  (ti), 
only.  Because  first,  third,  and  fifth  order  derivates  are  missing  from  this 
equation  the  associated  characteristic  equation  can  be  formulated  as  a 
cubic  algebraic  equation.  Again  it  is  found  that  for  the  range  of  the 


1476 


present  study  all  of  the  roots  are  real  Designating  these  roots^  as 

R  ,R,,  and  R3,  it  follows  that  four  solution  cases  are  possible  depending 

orl  the  coefficients  of  the  characteristic  equation.  They  are. 

Case!,  R, ,  R,,  and  R.  <  0-0  Case2,  R, ,  R^  <  0-0  ;  R3  >  0-0 

‘  -  (19) 

Cases,  Rj  <  0-0 ;  R,  and  R3  >  0-0  Case4,  Rp  R,  and  R3  >  0-0 


Inthepresentstudyonlycase3,andcase4are€ncountered.Introdudng  a=^\ Rj  1  , 

P  =  ,  a n d  y  =  JIR3 1  and  recognizing  that  (q ) must  be 

antisyrnmetric  about  the  ^  axis  while  X^(q)and  Y^(q)must  be 
symmetric,  we  are  able  to  write  for  case  4, 


Y^(q)  =  A,^  cosh  aq+B^^  cosh  Pq  +  C^^^  cosh  yq 


(20) 


Utilizing  the  coupling  of  the  ordinary  differential  equations,  as  in 
Reference  [4],  it  follows  that  we  may  write, 

X^/q)  =  A^R,^i  cosh  ocq  +  B^R^,  N  +  (21) 


and 

Z^(q)  =  A^S^,  cosh  ocq  +  B^^S^,  cosh  Pq+C,^iS^3  cosh  yq  (22) 

The  quantities  R^, , ,  S^p  ...  etc.,  are  evaluated  following  steps  described 
in  Reference  [4].  Expressions  for  (q ) ,  X^^  (q ) ,  etc.,  for  case  3  will  differ 
from  the  above  expressions  only  iii  that  Cosh  a  q  must  be  replaced  by  cos 
a  q. 

The  unknown  constants  A^ ,  B  „p  etc.,  of  the  above  solutions  are  evaluated 
by  enforcement  of  boundary  conditions  along  the  edge,  q=l.  These 
conditions  comprise  zero  torsional  moment,  zero  edge  rotation,  with 
transverse  shear  force  =  E^p  For  case  4  we  obtain 

(n)  =  —  { cosh  a  q  +  X 1  cosh  p  q  +  X  2  cosh  y  q }  (23) 


with  the  functions  X^(q)  and  (q )  differing  from  Y,^(q)  only  in  that 


1477 


R^j  and  S^j ,  etc.,  must  be  included. 

We  therefore  now  have  the  exact  response  of  the  first  building  block  to 
the  imposed  driving  force  components  available.  It  will  be  observed  in 
Figure  1  that  the  second,  fifth,  and  sixth  building  blocks  differ  from  the 
first  only  in  that  they  are  driven  along  different  edges.  Solutions  for  their 
response  are  therefore  easily  extracted  from  that  of  the  first. 

Focusing  our  attention  on  the  third  building  block  we  find  that  its  driven 
edge  is  free  of  transverse  shear  forces  and  torsional  moment.  It  is  driven 
by  a  distributed  cross-section  harmonic  rotation.  The  spacial  distribution 
of  this  imposed  driving  rotation  is  also  represented  by  the  series  of 
Equation  1. 

The  reader  will  appreciate  that  a  solution  for  the  third  building  block  is 
obtained  by  following  steps  identical  to  those  described  for  the  first.  Only 
the  imposed  boimdary  conditions  along  the  driven  edge  differ.  Solutions 
for  the  quantities  W,  etc.,  will  be  identical  in  form  to  those  already 
developed  for  the  first  building  block  except  that  quantities  XI,  XI,  etc., 
will  be  slightly  different.  We  designate  them  as  XIP,  X2P,  etc.,  for  the 
edge-rotation  driven  building  blocks.  Solutions  for  the  remaining  four 
building  blocks  of  Figure  1  are  therefore  available. 


THE  EIGENVALUE  MATRIX 

This  matrix  is  shown  schematicaly  in  Figure  2.  It  is  generated  following 
established  practices.  Let  us  first  consider  the  transverse  force 
equilibrium  condition  along  the  edge,  rj  =  1.  It  is  readily  shown  that  this 
equilibrium  condition  is  written  in  dimensionless  form  as, 

Q,  +  K„W  =  0  (24) 


The  plus  sign  of  this  equation  must  be  replaced  by  a  minus  sign  when  we 
formulate  the  corresponding  equations  for  the  edges,  11=0,  and  ^=0. 


1478 


Em 

1  2  3 

^  En 

1  2  3 

c:> 

CM 

C'>  - 

W4 

^  Ep 

i  2  3 

- 1  10 

^  E, 

1  2  3 

r&ean-, 
%  < 

^  Es 

1  2  3 

1  2  3 

- 

- 

-  -  ■ 

- 

::: 

- 

:  :  : 

LvJ 

:  :  : 

- 

-  -  - 

- 

_  -  - 

- 

:  :  : 

- 

- 

«  -  - 

- 

Ill 

- 

:  :  : 

- 

1 

1  M  1 

t 

p- 

-  ^  - 

-  -  - 

- 

:  :  : 

:  :  : 

- 

□ 

- 

:  :  : 

- 

- 

- 

:  :  : 

* _ 

[v 

+ 

r 

_  -  - 

- 

-  -  - 

- 

-  “  " 

- 

:  :  : 

■ 

- 

:  :  : 

- 

:  :  : 

Ill 

- 

:  :  : 

1  M 

1 

i" 

j 

]' 

:  :  : 

- 

:  :  : 

- 

"  "  I 

■ 

I  I  - 

Pig  2  Schematic  representation  of  Eigenvalue  matrix  based  on 

three-term  function  expansions.  Short  bars  indicate  non¬ 
zero  elements.  M  or  V  on  inserts  to  right  indicate  edges 
along  which  moment  or  lateral  force  equilibrium  is 
enforced. 

To  construct  the  first  three  equations  upon  which  this  matrix  is  based  we 
superimpose  all  eight  building  blocks  and  expand  their  net  contribution 
to  displacement  W  in  a  cosine  series.  The  transverse  shear  force  along  the 
edge,  11=1,  is  already  available  in  such  a  series.  We  then  express  the  left 
hand  side  of  equation  24  in  series  form  and  require  that  each  net 
coefficient  in  this  series  must  vanish.  This  leads  to  3  homogenous 
algebraic  equations  relating  the  8  k  imknowns  where,  for  the  illustrative 
matrix  of  Figure  2,  k  equals  3. 

A  second  set  of  three  homogenous  algebraic  equations  is  obtained  by 
enforcing  the  corresponding  lateral  equilibrium  condition  along  the  edge. 


1479 


^  =  1.  Moving  down  the  matrix  of  Figure  2  it  is  seen  that  a  third  and 
fourth  set  of  equations  are  obtained  by  enforcing  the  moment 
equilibrium  condition,  i.e.,  net  bending  moment  equals  zero  along  the 
same  edges,  in  an  identical  fashion.  Finally,  it  is  seen  in  Figure  2  that  four 
more  sets  of  equations  are  obtained  by  enforcing  the  required 
equilibrium  conditions  along  the  edges,  ti=0,  and  ^=0. 

We  thus  have,  in  general,  8  k  homogenous  algebraic  equations  relating 
the  8  k  unknown  driving  coefficients.  The  coefficient  matrix  of  this  total 
set  of  equations  forms  our  Eigenvalue  matrix. 

Certain  measures  can  be  taken  to  greatly  simplify  and  expedite 
generation  of  the  matrix.  It  will  be  observed  (Fig.  2)  that  the  matrix  is 
composed  of  64  natural  segments.  This  array  of  segments  may  be 
referred  to  through  the  indices  (I,  J).  It  is  expedient  to  first  generate  the 
matrix  without  including  contributions  related  to  the  driving  shear 
forces  along  the  building  block  edges  (Eqn.  24).  The  matrix  is  then 
completed  by  adding  the  quantity  1.0  to  diagonal  elements  of  segments 
(1,1)  and  (2,2),  and  subtracting  1.0  from  the  diagonal  elements  of 
segments  (5,5)  and  (6,6). 

Physical  reasoning  leads  also  to  another  vast  signification.  One  may 
begin  by  generating  the  elements  of  the  matrix  lying  below  the  first  four 
building  blocks,  only,  (Fig.  2).  Following  a  proceedure  as  discussed  in 
Reference  [4],  and  exercising  caution  with  respect  to  necessary  sign 
changes,  all  of  the  remaining  segments  of  the  matrix  may  be  extracted 
from  those  already  generated. 

Eigenvalues  are,  of  course,  those  values  of  the  dimensionless  frequency, 
X-,  which  cause  the  determinant  of  the  Eigenvalue  matrix  to  vanish. 
Mode  shapes  are  obtained  after  setting  one  of  the  non-zero  driving 
coefficients  equal  to  zero  and  solving  for  the  others. 


PRESENTATION  OF  COMPUTED  RESULTS 

It  will  be  appreciated  that  problems  involving  vast  arrays  of  stiffness 
coefficients,  plate  aspect  ratios,  thickness-to-length  ratios,  etc.  can  be 
resolved  by  the  proceedure  described  above.  Only  a  single  typical 
problem  and  its  solution  will  be  discussed  here,  for  illustrative  purposes. 

We  consider  a  square  plate  with  equal  dimensionless  lateral  elastic 


1480 


stiffness  imposed  along  each  edge.  Results  are  presented  for  two 
thickness-to-length  ratios,  0.01,  and  0.1.  Two  important  observations 
may  be  made  before  examining  these  results.  First,  for  the  very  thin  plate 
of  thickness-to-length  ratio,  0.01,  we  expect  the  Eigenvalue  vs  edge- 
stiffness  ratio  curves  to  almost  co-inside  with  those  for  a  thin  isotropic 
plate  based  on  thin  plate  theory.  Secondly,  we  recall  that  the  Eigenvalue 
limits  for  a  thin  plate  will  equal  0.0,  and  2  as  the  elastic  stiffnesse 
approaches  natural  limits  of  0.0,  and  infinity. 


Results  of  a  free  vibration  study  of  the  above  plate  are  presented  in 
Figure  3.  It  will  be  noted  that  computed  Eigenvalues  are  plotted  against 
the  parameter  Kli  /  (j)J.  By  presenting  data  in  this  manner  it  is  found 


1481 


that  Eigenvalues  for  both  the  thick  and  the  thin  plate  can  be  plotted  on 
the  same  Figure.  The  absissa  of  the  figure  appears  in  five  logarithmic 
decades.  This  range  has  been  selected  with  a  view  to  providing 
information  for  the  reader  over  the  region  of  greatest  interest.  It  is  found 
that  in  fact  the  thin-plate  curve  approaches  the  known  limits  disucssed 
above.  Furthermore,  utilizing  computing  schemes  related  to  Reference 
[1],  it  has  been  shown  that  this  same  curve  lies  extremely  close  to  that 
obtained  for  thin  plates  based  on  thin  plate  theory.  Because  Mindlin 
theory  takes  rotary  inertia  into  effect  and  does  not  consider  resistance  to 
transverse-shear  induced  deformation  to  be  infinite  the  thin  plate  curve 
of  Figure  3  will  lie  only  perceptably  below  that  of  its  companion  curve 
based  on  classical  thin  plate  theory.  The  lower  limit  for  the  thicker  plate 
curve  of  Figure  3  will  also  equal  zero.  However,  the  upper  limit,  equal 
to  that  of  a  simply  supported  thick  Mindlin  plate  will  lie  below  the 
classical  value  of  2n^.  This  is  seen  to  be  the  case  in  Figure  3. 


DISCUSSION  AND  CONCLUSIONS 

The  superposition  technique  is  seen  to  constitute  an  accurate,  straight¬ 
forward  technique  for  obtaining  analytical  type  solutions  to  the  problem 
of  analysing  free  vibration  of  Mindlin  plates  resting  on  uniform  lateral 
elastic  edge  support.  Convergence  is  found  to  be  rapid.  The  seven  terms 
utilized  here  in  representing  building  block  solutions  are  found  to  be 
more  than  sufficient  to  provide  four  digit  accuracy  in  computed 
Eigenvalues. 

The  reader  will  appreciate  that  the  analysis  described  here  could  easily 
be  modified  to  handle  plates  with  lateral  elastic  support  along  less  than 
four  edges.  Plates,  for  example,  with  two  adjacent  free  edges  and  the 
other  two  given  uniform  lateral  elastic  edge  support  can  easily  be 
handled.  It  is  only  necessary  to  set  the  elastic  stiffness  coefficients  equal 
to  zero  for  the  first  two  edges.  It  will  also  be  appreciated  that  through 
proper  choice  of  building  blocks  one  can  analyse  plates  with  one  or  more 
edges  resting  on  elastic  support  while  the  others  are  given  various 
combinations  of  simply  supported,  clamped,  or  free  boimdary  conditions. 
Natural  extensions  of  the  present  analysis  would  lead  to  the  solution  of 
problems  involving  the  same  plates  resting  on  arbitrarily  distributed 
elastic  edge  support  and  combinations  of  lateral  and  rotational  elastic 
support. 


1482 


LIST  OF  SYMBOLS 


a,  b 

D 

E 

G 

h 

ki,  k2,  etc.. 


Kur  Kl2,  etc., 


Plate  edge  dimensions 
-  E  hV(12  (1  -  )),  Plate  flexural  rigidity 

Youngs  modulus  of  plate  material 
Modulus  of  elasticity  in  shear  of  plate  material 
Plate  thickness 

Basic  lateral  spring  stiffness  along  plate  edge. 
Subscript  1  indicates  edge,  ti=1.  2, 3, 4,  indicate  edges 
moving  counter-clockwise  from  1. 

Dimensionless  lateral  elastic  edge  coefficients, 

k,  aw  k.,aw 

_! _  -  etc. 

K^-Gh  ’  K-Gh' 


M^,  M, 

w 


■y 

K" 

V 


V2 

V3 

4) 

4>h 


Dimensionless  bending  moments  associated  with  ^ 
and  r\  directions,  respectively 

Dimensionless  twisting  moment 

Dimensionless  shear  forces  associated  with  ^  and  r| 
directions,  respectively. 

Plate  lateral  displacement  divided  by  side  length  a 

Distances  along  plate  co-ordinate  axes  divided  by 
side  lengths  a,  and  b,  respectively 

Mindlin  shear  factor  =  0.8601 

Poisson  ratio  of  plate  material 

=  (l-v)/2 

=  (1  + v)/2 

=  6  (1  -  v) 

Plate  aspect  ratio  =  b/a 
Plate  thickness  ratio  =  h/a 


1483 


}\f^  Plate  cross-section  rotations  associated  with  ^  and  r\ 

directions,  respectively 

CO  Circular  frequency  of  plate  vibration 

=  CO  a “  ^p/D,  Free  Vibration  Eigenvalue 
p  Mass  of  plate  per  unit  area 


REFERENCES 

1)  Gorman,  DJ.,  “A  General  Solution  for  the  Free  Vibration  of 
Rectangular  Plates  with  Arbitrarily  Distributed  Lateral  and 
Rotational  Elastic  Edge  Support”,  Journal  of  Sound  and  Vibration, 
1994,  Vol.  (174),  No.  4, 451-459. 

2)  Saha,  K.  N.,  Kar,  R.C. ,  and  Datta,  P.K.,  “Free  Vibration  Analysis 
of  Rectangular  Mindlin  Plates  with  Elastic  Restraints  Uniformly 
Distributed  Along  the  Edges”,  Journal  of  Sound  and  Vibration,  1996, 
Vol.  (192),  No.  4, 885-904. 

3)  Gorman,  D.J.,  “Accurate  Free  Vibration  Analysis  of  Shear- 
Deformable  Plates  with  Torsional  Elastic  Edge  Support”,  to  be 
published  in  Journal  of  Sound  &  Vibration. 

4)  Gorman,  D.J.,  “Accurate  Free  Vibration  Analysis  of  the  Completely 
Free  Rectangular  Mindlin  Plate”,  Journal  of  Sound  and  Vibration, 
1996,  Vol.  (189),  No.  3, 341-353. 


1484 


Wave  Equation  Eigensolutions  on  Asymmetric  Domains 


R.  G.  Parker 

Department  of  Mechanical  Engineering 
Ohio  State  University 

C.  D.  Mote,  Jr. 

Vice  Chancellor-University  Relations,  FANUC  Chair  in  Mechanical  Systems 
Department  of  Mechanical  Engineering 
University  of  California,  Berkeley 


The  wave  equation 


Introduction 

-  V-^ -i- =/(x,y,r)  P 


q+^qn^O  dP 


(1) 


is  arguably  the  most  widely-studied  differential  equation  in  science  and  engineering.  It  is  used  to  model 
physical  systems  in  diverse  fields  such  as  acoustics,  wave  propagation,  vibration,  electromagnetics,  fluid 
mechanics,  heat  transfer,  and  diffusion.  Eigensolutions  of  the  two-dimensional  wave  equation,  governed 
by  the  Helmholtz  equation,  are  the  focus  of  this  paper.  We  present  a  perturbation  method  to  analytically 
calculate  eigensolutions  when  asymmetric  perturbations  are  present  in  the  boundary  conditioiis.  The 
axisymmetric,  annular  domain  case  serves  asjthe  unperturbed  problem.  Possible  boundary  condition  p^- 
turbations  include  deviation  of  the  domain  P  from  annular  and  variation  of  the  parameter  p  along  the 
boundary  dP .  The  eigensolution  perturbations  are  determined  exactly,  and  their  algebraic  simplicity 
allows  extension  of  the  perturbation  through  fifth  order.  Both  distinct  and  degenerate  eigenvalues  of  the 
unperturbed  problem  are  examined.  Boundary  condition  asymmetry  splits  the  degenerate  unperturbed 
eigenvalues.  We  derive  simple  rules  predicting  this  splitting  at  both  first  and  second  orders  of  perturbation. 
To  illustrate  the  method  and  quantify  its  accuracy,  the  case  of  domain  shape  perturbation  from  circular  is 
addressed  in  detail  and  comparisons  are  made  with  the  exact  solutions  for  elliptical  and  rectangular 

domains.  .  . 

Methods  used  to  analyze  eigensolutions  of  the  wave  equation  on  irregular  domains  have  been  pri¬ 
marily  numerical;  for  example  finite  element,  finite  difference  and  others.  In  addition  to  an  extensive  sum¬ 
mary  of  theoretical  results,  Kuttler  and  Sigillito  [1]  provide  a  comprehensive  review  (142  references)  of  the 
application  of  these  and  other  less  popular  methods.  Mazumdar  also  reviews  approximate  methods 
invoked  for  this  problem  [2,3,4].  The  above  methods  can  be  augmented  by  conformally  mapping  the 
irregular  domain  to  a  circle  [5].  In  the  spirit  of  perturbation,  Joseph  [6]  employed  a  parameter  differentia¬ 
tion  method  to  obtain  derivatives  of  the  distinct  eigenvalues  as  the  domain  changes.  The  requirement  of  a 
smooth  mapping  function  from  the  unperturbed  domain  to  the  irregular  one  and  the  restriction  to  distinct 
eigenvalues  of  the  unperturbed  problem  limit  its  applicability.  By  assuming  expressions  for  the  lines  of 
constant  deflection  in  the  fundamental  eigenfunction,  Mazumdar  obtained  estimates  for  the  fundamental 
eigenvalue  for  arbitrarily  shaped  domains  [7].  Accuracy  of  this  method  depends  on  the  availability  of  a 
good  estimate  of  the  lines  of  constant  deflection.  Morse  and  Feshbach  [8]  used  a  perturbation  analysis  dif¬ 
ferent  than  that  presented  herein  to  study  the  Helmholtz  equation  on  irregular  domains.  Expansion  of  the 
eigenfunction  perturbations  in  infinite  series  of  the  unperturbed  eigenfunctions  leads  to  a  convergence 
problem  restricting  the  analysis  to  second  order  perturbation  in  the  eigenvalue  and  first  order  in  ^e  eigen¬ 
function.  Nayfeh  used  a  perturbation  formulation  similar  to  that  of  this  work  to  calculate  the  eigenvalue 
perturbation  to  first  order;  no  eigenfunction  perturbations  are  presented  [9].  This  work  draws  on  the  results 
of  Parker  and  Mote  [10, 11],  where  a  formal  procedure  for  obtaining  exact  eigensolution  perturbations  is 
developed. 


1485 


Eigensolution  Perturbation  Formulation 

The  eigenvalue  problem  .resulting  from  separation  of  the  spatial  and  temporal  dependence  in  (1)  by 
the  assumption  q  =/?(/?,  0)e is 

-  VV  -  =  0  P  (2a) 

p  +  =  0  :  dPi  [j  dP(,  (2b) 

where  subscript  n  denotes  the  normal  derivative.  We  examine  two  classes  of  asymmetry  that_normally 
preclude  exact  determination  of  the  exact  solution  to  (2):  irregular  P  and  variation  of  P  along  dP .  These 
asymmetries  are  treated  as  perturbations  of  the  axisymmetric,  annular  domain  eigenvalue  ^oblem. 

(i)  Irregular  Domain  Shape:  The  two-dimensional,  doubly-connected  domain  P  in  Fig.  1  is 
P:  Ri(Q)<R  <Ro(Q),0<Q<  2%.  The  deviations  of  the  boundaries  dPi  and  dPg  from  circular  are 

£^.(0)^,  (0)  ■  Ri  (6)  -  Ro  (3) 


where  Ri  and  R^  are  the  average  radii  of  the  inner  and  outer  boundaries.  The  variables  q,p,  and  t  of  (1) 
and  (2)  are  dimensionless;  scaling  the  domain  with  the  additional  dimensionless  variables 


r  =  -§-  Y=^  (a  =  R,a  eg,(9) 

Pq  Po 

Ego(Q) 

Ro 

- 

Pq 

(4) 

yields  the  eigenvalue  problem  for  constant  P  =  Po 

-  V^p  -  (£>^p  =  0 

P 

(5a) 

p  +^oPn=0 

3? 

(5b) 

where?:  Y-i-  £g,(0)<r  <1  +  g^^(0),  O<0<27t. 

Boundary  quantities  on  dP  are  approximated  by  Taylor  series  expansion  about  r  =  y,  1.  For  exam- 
ple, 

?  lr=I+£g,  =  ?  Ir=l  +  i^8o)Pr  I  r=l  +  Prr  \  r=l  +  Prrr  1  r=I  +  ‘  '  (6) 

A  similar  expansion  is  developed  for  p  ]  3^, .  The  expansions  |  require  asymptotic  expansion  of  p,i 
in  terms  of  derivatiyes  with  respect  to  the  polar  coordinates  r  and  ^  [12].  Introduction  of  the  expansions 
for  p  and  p^  on  dP  into  (5b)  yields 

-  =  0  P:  Y<r  <1,  O<0<27t  (7a) 

(? -Po?r)  +  eQ?  +  ■  ■  ■  =0  r=y  (7b) 

ip  -f  Po  Pr)  +  zCp  +  Z^Dp  +z^Ep  +  •  •  •  =  0  r  =  l  (7c) 

where  C ,D ,C ,D ,  ‘  •  are  linear  boundary  operators  with  variable  coefficients  depending  on  (0)  and 
8 

(ii)  Variable  p  along  dP :  For  annular  domains  with  P  — >  P^,  +  E  p(0),  the  eigenvalue  problem  becomes 

-W^p -a^p  =0  ?:Y<r  <l,O<0<27t  (8a) 

(?  -  Po?r)-£p(0)?r  =(?  -  PoPr) +  =0  ^=7  (8b) 

(?  +  PoPr)  +  eP(0)?r  =(?  +  PoPr) +  =  0  '‘  =  1  (8c) 

which  is  of  form  identical  to  (7). 

We  seek  eigensolutions  of  (7)  where  the  boundary  perturbations  may  result  from  either  or  both  of  the 
asymmetries  discussed  above.  The  eigensolutions  are  represented  as  asymptotic  series  in  the  small  param¬ 
eter  £ 


1486 


03^  =  03^  +  ep.  +  e^T|  +  +  +  O(s^) 


(9) 


p=u  +  EV  +  z^w  +z^s  +£^r  +eh  +0(£^)  (10) 

Subsequent  analysis  shows  that  confinement  of  the  perturbation  terms  to  the  boundary  conditions  ensures 
that  the  Bessel  and  trigonometric  forms  of  the  eigenfunction  perturbations  in  (10)  do  not  depend  on  the 
boundary  perturbations.  Only  the  coefficients  of  these  Bessel  and  trigonometric  functions  depend  on  the 
boundary  condition  operators.  Because  of  this  essential  point,  the  method  presented  in  the  sequel  for 
finding  exact  solutions  for  the  eigensolution  perturbations  on  irregular  domains  can  be  readily  applied  to 
find  exact  eigensolution  perturbations  for  any  problem  of  the  form  (7). 

With  the  inner  product  <e  ,f  >  dA,\ht  normalization  <p  ,p>  =  1  and  (10)  give 

P 


<u,v>  =  0  <u,w>  =  -V2<v,v>  <u,s>=-<v,w> 

<u,t>  =  ~  <v,s>  -  V2< w ,  w  > 


(11) 


The  eigenvalue  perturbations  li,T|,  K,X,  and  G  are  determined  subsequently  in  terms  of  the  boundary 
conjuncty  (£ ,  / ) 

J(e,f)  =  <  -Vhj>-<e.  -VV>=  j  [e/„  ■fe„]ds-  (12) 

dP 


Irregular  Domain  Eigensolution  Perturbation 

We  demonstrate  the  solution  procedure  for-ei  gen  value  problems  of  the  form  (7)  by  examining  an 
irregularly-shaped  domain  with  outer  boundary  dP  and  boundary  condition  p  1  gp  -  0.  Use  of  (4)  and  (6) 
yields 

-V^p-'^^p=0  P:0<r  <1,O<0<27C  (13a) 


p  +Zgp,+  1/lKtgfprr  +  \m^gfpnr  +  WK^gtPrrrr  +  l/5!(£«  =0  r  =  1  (13b) 

where  the  subscript  o  denoting  the  outer  boundary  has  been  omitted.  Substitution  of  (9,  10)  into  (13) 
yields  the  sequence  of  perturbation  problems 


-  M  -  03^  M  =  0  P 

(14a) 

U  =  0  dP 

(14b) 

-  V  -  03^  V  =  jl  W  P 

(15a) 

V  ^  -g  Ur  dP 

(15b) 

-V“W  -03~W  =  |iV  -I-TIM 

P 

(16a) 

W  =  -gV,  ~{m)g\r 

dP 

(16b) 

-05^  5  =  |IW  -fTlV  -l-KW 

P 

(17a) 

5  =  -  gw r  -  (1/2)  g\rr  *  8^^rrr 

dP 

(17b) 

-V^r-03^r  -\is  +r\w  +  kv  +  X“ 

P 

(18a) 

t  =  -gSr  -  (l/2)g  Vr  -  (l/6)5\rr  " 

{1/24)  g\rrr 

(18b) 

1487 


-V^z  -CO^Z  =|lf  +T15  -1-KW  +XV  +GU 

p 

(19a) 

z  -  -  gt,-  -  (l/6)g  Vrr  -  (l/24)g  Vrrr 

-(l/120)gVrrr 

(19b) 

Solution  of  (14)  gives  the  orthonormal  unperturbed  eigenfunctions 

1 

^mO  ~  lA  T  f  r,\  \ 

m>0 

(20) 

cs  2  >^0  ( ®/nn )  [cos  n0  d  /  ^\l 

cos  n  0  -..n  A 

sinne  '«^0,n>0 

(21) 

where  m  and  n  denote  the  number  of  nodal  circles  and  nodal  diameters  in  the  eigenfunction.  The  unper¬ 
turbed  eigenvalue  is  the  (m  +  1)'"  root  of  the  characteristic  equation  (CO)  =  0.  C0^„  is  a  distinct 
eigenvalue  for  «  =0  and  a  degenerate  eigenvalue  of  multiplicity  two  for  n  >  1.  are  orthonormal 
eigenfunctions  associated  with  the  degenerate  eigenvalues. 

For  the  circular  domain  P ,  the  boundary  conjunct  (12)  is 


f  [efr -ferUidQ 

0 


The  following  relations  are  used  subsequently 


o'  n'*- 

“  '  ^i/j 


K 


Rmna)=  - 


r  1 
2 

V2 

2 

% 

^mn  ■^7««(1)  ~ 

K 

g  ( ©)  =  X  cosy  0  +  X  sin;  0 

;=i  7=1 


(22) 

/z 

C0^,i:23) 

(24) 

(25) 


G(0)  =  g2(0)  =  Go  +  X  G/cos;0  +  X  OjsinjQ 

7=1  7=1 

Use  of  the  Fourier  representation  (24)  allows  treatment  of  arbitrary  boundary  shapes,  including  kinked  or 
discontinuous  boundaries.  The  constant  term  in  (24)  vanishes  because  R  is  the  average  radius  of  the  boun¬ 
dary. 


Solution  of  Perturbation  Equations 

Distinct  Eigenvalue  Perturbation 

Consider  perturbation  of  a  distinct  unperturbed  eigensolution  (CO;„o>  ^mo)  (lii®  subscript  mO  will  be 
omitted  in  the  sequel).  Using  solvability  conditions  for  the  perturbation  problems  (15-19),  Parker  and  Mote 
present  formal  expressions  for  the  distinct  eigenvalue  perturbations  in  terms  of  the  boundary  conjunct 
[10,11] 

p.  =  -J{u,v)  Tl  =  -Jiu,w)  K=  -\i<u,w  >  -  J(^u,s )  (26) 


X=  -  \1<U,S>  -T\<U,W>  ‘  J(u,t)  a=  -\L<U,t>  -r\<U,S>  ~K<U,W>  -J(u,z) 

1st  Order  Perturbation:  The  first  order  eigenvalue  perturbation  }I  is  evaluated  from  (26a),  (22),  (14b), 
(15b),  (20),  and  (24) 

2tc  2n 

0  0 

The  result  p.  =  0  substantially  simplifies  subsequent  calculations.  It  results  because  the  radius  of  the  unper¬ 
turbed  circular  domain  is  the  mean  radius  of  the  irregular  domain. 


1488 


In  their  treatment  of  boundary  shape  perturbation  of  the  Helmholtz  equation,  Morse  and  Feshbach 
[8]  noted  convergence  difficulties  when  the  eigenfunction  perturbation  is  expanded  in  a  series  of  the  unper¬ 
turbed  eigenfunctions.  For  the  similar  case  of  plate  boundary  shape  perturbation,  Parker  and  Mote  [12] 
encountered  a  divergent  series  for  the  second  order  eigenvalue  perturbation  when  the  first  order  eigenfunc¬ 
tion  perturbation  is  expanded  in  a  series  of  the  unperturbed  eigenfunctions.  These  problems  do  not  occur 
when  the  exact  solution  for  the  eigenfunction  perturbation  is  determined.  Furthemore,  the  exact  solufion 
is  more  accurate,  computationally  efficient,  and  notationally  conveiiient  than  the  infinite  series  expansion. 
In  the  sequel,  exact  eigenfunction  perturbation  solutions  are  determined  through  fourth  order  perturbation, 
thereby  allowing  exact  calculation  of  the  fifth  order  eigenvalue  perturbation. 

The  eigenfunction  perturbation  v  (  r ,  9)  is  decomposed  as 

V  =  c  «  -h  (28) 

The  first  term  results  because  w  is  a  non-trivial  solution  of  the  homogeneous  form  of  (15a,b);  c  is  a  con¬ 
stant  to  be  determined.  The  second  term  is  the  general  solution  of  the  homogeneous  form  of  (15a).  The 
third  term  of  (28)  is  a  particular  solution  of  the  inhomogeneous  equation  (15a),  Because  of  (27), 

=  0  (29) 


Additionally, 


=  £y;(cor)[i5y  cosjQ  +  Cj  sinyG]  (30) 

where  the  j  =  0  term  is  omitted  because  its  contribution  is  included  in  the  first  term  of  (28).  The 
coefficients  in  (30)  are  calculated  from  (15b) 

^  _  ®g/ 


rVy(CO) 


=  I/. 

^  7t^Jj(C0) 


(31) 


(32) 


where  and  are  from  (24).  Substitution  of  (28)  into  the  normalization  condition  (11a)  yields 

c  -  -  <M,  v^'  -f  v^>  =  0 

and  the  solution  for  v  is  complete. 

2nd  Order  Perturbation:  The  second  order  eigenvalue  perturbation  T|  is  evaluated  from  (26b),  (22), 
(14b),  (16b),  (20),  (28-32),  (24),  and  (25) 

Tl  =  j  [Mr  (  -  g^  -  V2g\r)'\r^idQ  =  ^0+1.  ^ 


The  solution  to  (16)  is  decomposed  as 

w  =  d  u  + 

where  the  definitions  of  the  terms  in  (34)  are  analogous  to  those  in  (28),  and 

r\r  d  t(cor) 


vP  =  - 


(47t)‘'^C0  ^i(co) 


(34) 

(35) 

(36) 


w  -'Z  dj  (wr )  [  Ej  cos;  6  -i-  Fj  sin  j  0  ] 

j=i 

The  coefficients  Ej  and  Fj  in  (36)  are  determined  from  (16b) 

^  gm(gm+;  8m-j^  8m^8m+J  8m~j^  ^  ^7 

r-’C  r  dfji+ii(ii) 

Gf+  X  -ft)  , . . 3 


£,=  - 


(47:)'^Vy(0)) 


4  (CO) 


8m^8m+j  -^gy-m)  g»i(gm+;  "gy-m)  ^ 

m  =; 


8m  8  2m  8m  8  2m 


1489 


Fj=. 


CO 


From  the  normalization  (11b), 


^  r  T  J 


8m^8m+j  ~  8m-j^  ~  8m^8m+j  ~  8m-j^ 
8m^8m-^i  8j-m)  ~  8m(8m+j  ~  8j~m) 

[_  8m  8  2m  ~  8m  8  2m 


1- 


d  =  -V2<v,v>-<u,wP> 
’m^l  (CO) 


2m 


■  + 


CO  ^(CO)  y^(co) 


m  >j 
m  <j 
m  -  j 

(37) 


■  s  ■\2n 


T1 


=  'T  ^ 

^  m=l 

1. 

The  particular  solution  (35)  is  the  critical  component  of  the  solution  (34).  With  known,  calculation  of 
E;  and  Ft  is  straightforward  for  any  perturbed  boundary  conditions  (7b,c).  ,  ■  r  • 

^  Equations  (27-37)  provide  exact,  closed-form  expressions  for  both  the  eigenvalue  and  eigenfuncuon 
perturbations  through  second  order.  Their  simplicity  is  remarkable  given  that  they  apply  for  an  arbitrary 
deviation  in  boundary  shape.  .  ^  • 

3rd  Order  Perturbation:  The  third  order  eigenvalue  perturbation  (26c)  is 

27t 


K  =  -  /(  M  ,  5 )  =  J  [  '  (1/6)  )lr=l  ^ 0 


(38) 


Derivation  of  a  closed-form  expression  for  K  is  analogous  to  (33)  and  straightforward.  The  value  of  the 
expression  is  minimal,  however,  because  (38)  can  be  evaluated  easily  using  computer  algebra  software  for 
a  specified  g  (6).  Little  insight  can  be  gained  from  the  algebraic  details  at  third  order  perturbation. 

The  third  order  eigenfunction  perturbation  ^  (  r ,  0)  is 

s  =  e  u  +  (39) 

A  particular  solution  for  the  Kw  term  in  (17a)  is  known  by  analogy  with  (35) 

(40) 


(47C)''^^CD  j  i(CJ)) 

A  particular  solution  associated  with  the  riv  term  of  (17a)  is 

-  TT  £  cos;e  +  Cj  sin/e] 


Finally, 


5P  = 


(41) 


(42) 


As  in  (30)  and  (36), 

=  £yy(cor)[//jCOs;0  +  Ly  siny0]  (43) 

M 

where  numerical  evaluation  of  Hj  and  Lj  for  specified  g  (0)  is  readily  achieved,  e  in  (39)  is  found  from 
(11c) 

e  =  -<v,w>  -<u,s^>  (^^4) 

4th  Order  Perturbation:  The  fourth  order  eigenvalue  perturbation  %  is  found  from  (26d) 

X=  -T\d  +  (45) 


1490 


The  fourth  order  eigenfunction  perturbation  ?  ( r ,  0)  is 

t  =  f  u  +  t''  +  tP 

Expansion  of  (18a)  yields 

-V^t  -oy^t  ^{x  +  T]d)u  +Kv^'  +T\w^'  +r\wP 
A  particular  solution  of  (47)  is 

tP  =ti  +t^+ti+  fP^ 

(X  +  ^d)  r 

(Anf^w  J  i(t^) 


- 


=  - 


Kr 


2co,tl 


£  Jj^ii(£>r)[Bj  cosy  e  +  Cj  siny  0] 


1?  =  -  ^  i;  Jj^iiar)  [  Ej  oos  j  6  +  Fj  sin;  6] 


(46) 

(47) 

(48) 

(49) 

(50) 

(51) 


tP.  =  - 


jfr 


[cor/oCwr)  -  2^1(0))] 


(52) 


is  the  only  ’new’  particular  solution  not  determined  by  analogy  with  previous  particular  solutions. 

Jr  .  •J.  Ji  1  1  .  j  /- _  /lOL'v  TT _ /I 


Coefficients  of  are  calculated  from  (18b).  The  normalization  (1  Id)  gives 
f  =  -  V2<  w,w>-<v,s>~<u,tP> 

5th  Order  Perturbation:  From  (26e),  the  fifth  order  eigenvalue  perturbation  a  is 


a  =  -  Tie  -Kd  +  -J{u,z) 


(53) 

(54) 


Degenerate  Eigenvalue  Perturbation  j  j  i  j- 

Consider  perturbation  of  a  degenerate  unperturbed  eigenvalue  C0^„  and  the  associated  n  nodal  diam¬ 
eter  orthonormal  eigenfunctions  (21)  (the  subscript  mn  will  henceforth  be  omitted).  Because  of  the 
eigenvalue  degeneracy,  the  unperturbed  eigenftinction  u  is  an  element  in  the  linear  space  spanned  by  u 
and 

u-a^u^+a^u^  (55) 

Cc  and are  determined  subsequently.  The  normalization  <u,u>  ~  1  requires 

+  (56) 

Boundary  condition  asymmetry  splits  the  degenerate  unperturbed  eigenvalue  and  fixes  the  coefficients  <3^ 
and  in  (55).  These  effects  might  occur  at  first  order  perturbation,  though,  if  not,  they  are  predicted  at 
some  higher-order  perturbation.  ^  j- • 

Because  and  are  solutions  of  the  homogeneous  forms  of  (15-19),  two  solvability  conditions 
must  be  satisfied  at  each  order  of  perturbation.  We  follow  the  method  of  Parker  and  Mote  [10]. 

1st  Order  Perturbation:  The  solvability  conditions  for  (15)  yield 

£2^11=  -/(mSv)  £2^  |1=  -y(M^v)  (57) 

Evaluation  of  (57)  yields  a  symmetric,  algebraic  eigenvalue  problem 


1491 


-  0)^ 


Sin  Sin 
Sin  ~  Sin 


=  [1 


Da  =  |xa 


\il,l=^±^His^^r^(sLf} 


l-iVz 


(58) 


The  eigenvalues  of  D  are  the  first  order  perturbations  of  ©.  The  eigenvectors  of  D  fix  the  coefficients  in 
(55).  The  degenerate  eigenvalue  splits  into  distinct  eigenvalues  as  a  result  of  boundary  asymmetry  if  and 
only  if^the  fi  are  ^stinct.  The  magnitude  of  jl  for  an  n  nodal  diameter  eigenvalue  is  proportional  to 
[  is  In)  +  iSin^i-  This  leads  to  the  splitting  rule:  If  either  or  both  of  g2n  and  g2n  are  nonzero,  the  n 
nodal  diameter  eigenvalues  split  at  first  order  perturbation;  otherwise  the  eigenvalues  remain  degenerate. 
When  no  first  order  splitting  occurs,  =  0  but  and  remain  undetermined. 

The  eigenfunction  perturbation  v  ( r,  6)  is  decomposed  as 

V  =  Cj.  +  Cj  «■*'  +  V*  +  (59) 

where  the  first  two  terms  result  from  the  two  independent  solutions  of  the  homogeneous  form  of  (15).  Par¬ 
ticular  and  homogeneous  solutions  of  (15a)  are 


4+l(C0r) 


(27r)'/^co  ^+i(CD) 


( a^.  cos nQ  +  sin n 0) 


(60) 


2  7y (cor)  [By  cos ;  9  +  Cy  sin j 0 ] 

J=0J*n 

where  the  j  =  n  term  of  (61)  is  included  in  the  first  terms  of  (59).  Using  (15b), 


(61) 


Bj- 


Cj  = 


CO 


(27t)^Jy(CO) 


(2n)%(cy) 


^dSj-^n  Sj~n)  ^s(Sj+n  ~  Sj-n) 

j>n 

^ciSj+n  Sn~d  ^siSj-i-n  ■*"  8n-j) 

j<n 

(62a) 

Sn  +  Sn 

o 

II 

•'"J 

^cisUn  +Sj-n)-^s(sUn  'Sj-n) 

j>n 

(62b) 

^ciSj+n  ~  Sn—d  ~  ^siSj-^-n  ~  Sn—j) 

j  <n 

V. 

Coefficients  c^.  and  completing  the  solution  (59)  are  calculated  at  second  order  perturbation,  just  as 
and  of  (55)  are  calculated  at  first  order  perturbation. 

2nd  Order  Perturbation:  The  two  solvability  conditions  for  (16)  and  (11a)  yield 


iH^Sin 

Cc 

ara,  0 

Tl/OO^ 

+  1) 
lap- 

|X^(n  +  1) 

203^ 

Vijn  +  1) 

2 


~J{u^ ,w) 
-  f{u\w) 


(63) 


1492 


(0^ 


+  a J  -  CO^C  ‘yiGi,)  £  u-  Pj  j-  =  a,  X  +  a,  y 


jj+m . 


j^.i*n 


•/;(C0) 


j=0J^n 


y,cco) 


(64a) 


•'wW. 


,2  OO 


^  j^J*n 


/;(C0) 


(64b) 


(gUn  +  Sj-nf  +  igj+n  +  Sj-nY 

P;  j 

~gj+ngj-n  gj-ngj+n 

i>n 

aj  =< 

(gj+n  +  8n~jY  +  (.gUn  -  gn-jY 

gj+ngn-j  gn-jgj+n 

j<n 

[2{sD- 

gngn 

j=o 

igj+n  ~  gj-n^  igj+n  ~  gj~n^ 

gj+ngj-n  gj+ngj-n 

j>n 

igUn  -  Sn-jY  +  (g/+„  +  gLjY 

J>} 

il 

gj+ngn-j  ~  gj+ngn-j 

j<n 

'lig^nY 

j=0 

6,-  =  (l/4)(  tty  -  Tj )  is  used  in  (66).  The  operator  in  (63)  is  invertible  if  and  only  if  the  unperturbed  eigen¬ 
value  splits  at  first  order  perturbation;  c^,  c^,  and  Tj  are  calculable  from  (63).  If  g2n  =  gin  ~ 

|J.  =  0,  the  operator  in  (63)  is  singular,  and  a^.  and  are  unknown.  In  this  case,  the  component  equations 
of  (63)  yield 


f{  u\w)  -  J{u^ ,w)  =  Y  (a^.^  -  +  (Z  -  X)a(.  <2^=0 

Ti  =  -  f{u^ ,w)  -  f(u\w)  =  -X  -2Y  -Z 


(65) 


where  X,  7,  and  Z  are  defined  in  (64).  The  first  of  (65)  and  (56)  can  be  solved  for  two  unique 
&-  {Uf.  a^Y  if  and  only  if  one  or  both  of  the  following  inequalities  hold 

y^o-4‘/4GL+  X 

j=OJ^n  -gx 

X  ;.z  ^  yaGi,  +  E  [;•  -  6y  ^0 

j=0J*n 

Ti  is  then  calculated  from  (65b).  Equations  (66)  are  second  order  eigenvalue  splitting  rules:  If  either  or 
both  of  (66)  are  satisfied,  the  n  nodal  diameter  eigenvalues  split  at  second  order;  otherwise  they  do  not. 
When  7  =  0  and  X  =  Z,  (65b)  and  (56)  yield  r\=  -  X  =  -  Z  while  (65a)  is  identically  satisfied.  Thus, 
r\  is  calculable  despite  the  lack  of  second  order  splitting  and  the  continuing  indeterminacy  of 
sequel,  we  assume  the  degenerate  eigenvalues  split  at  first  order.  If  they  do  not,  the  development  described 
by  Parker  and  Mote  [11]  is  required. 

The  second  order  eigenfunction  perturbation  w  ( r ,  6)  is 

w  =  dc  +  vv''  +  wP  (67) 


1493 


wf  =  - - - ; — — [  (M-c^  +  ria^ )  cos «  e  +  (|a.c^  +  '{\a, )  sin  n  6]  (68) 

(27U)^^C0 

=  i  y,.^.,(0)r)[B;Cosye  +  Cysin;e]  (69) 

w?  =  - - - [(HrJ-im)  -2(n  +  l)/„+i(cor)](ac  cosne  +  sinnO)  (70) 

(327c)/^coV„+i(co) 

=  wf  +>1^1+1^^  (71) 

w''  =  X  J;(©r)[£y  cos;e  +  F;  sinjO]  (72) 

j=0J^n 


Components  wf  are  associated  with  the  three  inhomogeneities  of  (16a)  resulting  from  an  expansion  analo¬ 
gous  to  (47).  The  particular  solution  (71)  is  the  essential  element  allowing  calculation  of  the  E;  and  F; 
from  (16b). 

3rd  Order  Perturbation:  The  solvability  conditions  for  (17)  and  (lib)  give 


(1  + 

ay^ac 

r  " 

4 

-\i<u'^,W^>  -T\<U*^  ,V>  -J(u^  ,s) 

<^Si, 

(0^a_y 

4 

h  ^  M 

-  \i<u\wP> -r\<u^  ,v>  -  f(u\s)  (738 

0 

k/co^ 

-  C0“[y2<V,V>  +  Oc  <U^  ,W^>  +  <M'^,  W^>] 

The  operator  in  (73)  is  identical  to  that  in  (63),  and  the  assumption  of  first  order  eigenvalue  splitting 
ensures  its  invertibility  in  the  calculation  of  d^yd^,  and  K. 

The  third  order  eigenfunction  perturbation  j  (  r ,  0)  is 

s  +esU^  +  5^'  +  sP  (74) 

f  /ri  +  l(d5l‘) 

^  - — -—-[{\id^  +T]Cc  +Ka^)cosnQ  +  {lLd,  +  Tic^  4-K^2jsinn0]  (75) 

(271)  ^+i(0)) 

X  COS70  4-  Cj  siny'e]  (76) 

- [cory„(C0r)  -  2(n  +  lV„+i(®r)](ac  cosnO  +  a,  sinnO)  (77) 

(327t)/W„.,i(co) 


(78) 

(79) 

(80) 


1494 


_ - +  ?) - [r/  (cor)  +  +l)_j  (a)r)](a^  cosnS  +  a,  sinnO)  (81) 

(288jc)’Vy„^.,((B)  2“ 


sP  =  S^l  +  +  >5'^ 


(82) 


s'^  =  £  7;(COr)[H,cos;0  +  L;sin;0] 

j=0,j*n 


(83) 


where  the  Hj ,  L;  follow  from  (17b). 

4th  Order  Perturbation:  From  the  solvability  conditions  for  (18)  and  (1  Ic) 


(irgln 

- 

r 

0 

X/w^ 

-  -T|<mSw>  -  k<m‘^,v>  -/(m^,0 

-  -rl<M^W>  -  K<W^V>  -  /(M^^) 
-(i?{<v,w>  +  ac<u\sP>-^  as<u\sP>], 


(84) 


The  presented  boundary  perturbation  method  applies  for  general  boundary  conditions  of  the  form 
(7b, c).  For  annular  domains,  the  Bessel  function  y„  (C0^„r)  is  included  in  (20,  21);  particular  solutions 
associated  with  this  additional  term  are  almost  identical  to  those  associated  with  /„(co^nt- )  [10].  If  dif¬ 
ferent  unperturbed  boundary  conditions  are  considered,  the  unperturbed  eigenfunctions  (20,  21)  do  not 
change  form;  only  the  normalization  coefficients  change.  Consequently,  the  form  of  the  right-hand  side  of 
(15a)  is  unchanged,  and  the  particular  solutions  (29,  60)  apply  £xcept  for  a  change  in  their  leading 
coefficients.  Different  first  order  boundary  condition  perturbations  C  and  C  change  only  the  values  of  the 
coefficients  Bj  and  Cj  in  (30,  61).  Instead  of  (15b),  these  coefficients  are  determined  by  the  general  per¬ 
turbed  boundary  condition  (7c) 

[  v"  +  yl!]3P  =  -  Ck  -  [  v'’  +  P,,  v?hp  (85) 


Calculation  of  B;  and  C;  is  always  possible  by  Fourier  expansion  of  (85).  Thus,  the  forms  of  v  in  (28) 
and  (59)  are  unaffected  by  changes  in  either  the  perturbed  or  unperturbed  boundary  operators.  As  a  result, 
only  coefficients  of  the  particular  solutions  (35,  71)  change  for  different  bound^  conditions.  This  reason¬ 
ing  extends  to  higher  order  perturbations,  and  consequently  the  prepnted  particular  solutions  admit  exact 
eigensolution  perturbations  for  general  boundary  condition  perturbations. 


Example  Problems 

The  numerical  accuracy  achievable  by  the  presented  method  is  illustrated  by  modeling  elliptical  and 
rectangular  domains  with  a  circle. 

Elliptical  Domain  ,  , , 

An  elliptical  domain  of  eccentricity  e  =  {I  •  is  described  by 

'  ,  V/2 

1  -  e- 


R  -  a 


1  -  e^cos^0 


(86) 


where  a  and  b  are  the  semi-major  and  semi-minor  axes,  respectively  (Fig.  2).  The.  average  radius  R  (3b) 
and  the  Fourier  coefficients  of  g  (0)  (24)  are  calculated  by  quadrature.  Though  R  depends  on  a  and  e , 
g(0)  depends  only  on  e .  For  an  ellipse,  g/  =  0  for  j  odd  and  g/  =  0  for  all  j .  Eight  non-trivial  terms 
through  g  16  were  used  in  the  calculations. 

The  dimensionless,  fundamental,  elliptical  domain  eigenvalue  (2)  evolves  from  the  funda¬ 

mental  circular  domain  eigenvalue.  (We  identify  the  perturbed  domain  eigensolutions  using  subscripts  mn 
denoting  the  number  of  nodal  circles  m  and  nodal  diameters  n  in  the  circular  domain  eigensolutions  firom 
which  the  perturbed  eigensolutions  evolve.)  Table  1  compares  the  fundamental  eigenvalue  predicted  by 


1495 


perturbation  to  the  exact  values  computed  by  Daymond  [13].  A  maximum  error  of  0.32%  is  calculated  for 
eccentricities  through  e  =  0.9 b/a  -  0.4359.  For  the  extreme  eccentricity 
e  -  0.961  \-^b/a  -  0.28,  the  perturbation  results  degrade  substantially.  Even  without  comparison  with 
a  known  solution,  the  degradation  is  evident  by  the  poor  convergence  of  the  perturbation  with  increasing 
order.  For  e  =  0.961 1,  llie  asymptotic  expansion  (9)  for  the  fundamental  frequency  is 

©2  =  5.7831  +  0  +  5.1154  -  5.3966  +  4.7782  +  1.7258  (Qqo^)  =  =  7.2790 

Ra=\  0.47603 

In  contrast,  for  €  =  0.9  _ 

=  5.7831  +  0  +  1.6774  -  0.6083  +  0.3098  -  0.0005  (floco  )  =  -f-  =  lUtu  ^ 

The  agreement  between  perturbation  and  the  exact  values  is  illustrated  in  Fig.  3.  Results  for  perturbation 
of  the  one  nodal  diameter  circular  domain  eigenvalue  are  also  given  in  Fig.  3,  where  the  exact  values  were 
obtained  by  optically  scanning  Fig.  1  of  Troesch  and  Troesch  [14]  and  digitizing  points  through  e  =  0.9. 
Differences  between  the  results  so  obtained  and  the  perturbation  values  are  all  less  than  3%,  which  is 
approximately  the  precision  of  the  scanned  and  digitized  results.  Tabular  results  for  the  (i2oi^)l,2  eigsR" 
values  are  presented  in  Table  1. 

Rectangular  Domain 

Consider  the  rectangular  domain  of  dimension  2a  x2b  where  q  =  b/a-<  1  (Fig.  4).  The  average 
radius  R  and  the  Fourier  coefficients  of  $  (0)  are  calculated  by  quadrature.  R  depends  on  a  and  q,  but 
g  (6)  depends  only  on  Also,  gj  =  0  for  j  odd  and  g/  =  0  for  ail  j .  Ten  non-trivial  terms  through  g20 
were  used  in  the  calculations  (Fig.  4). 

Table  2  compares  the  fundamental  eigenvalue  perturbation  to  the  exact  value  for 

1  >0.3.  Comparisons  are  shown  for  first  through  fifth  order  perturbation  approximations.  For  a  fifth 

order  perturbation  approximation,  errors  in  the  fiindamental  frequency  are  less  than  0.5%  for  ^>0.6,  1.3% 
for  ^  =  0.5,  and  4.7%  for  ^  =  0.4.  For  ^  =  0.3,  the  error  is  22.%  and  perturbation  is  not  effective.  The 
behavior  of  the  asymptotic  approximation  (vertical  column  of  Table  2)  reveals  a  large  expected  error  even 
in  the  absence  of  a  known  solution. 

Substantial  improvement  in  the  predicted  fundamental  eigenvalue  results  when  the  perturbation  is 
extended  from  first  to  second  order.  The  accuracy  obtainable  from  a  second  order  perturbation  is 
significant  because  the  closed-form  expression  (33)  gives  Tj  for  an  arbitrary  shape  perturbation.  Improved 
accuracies  are  achieved  with  third  order  and  fourth  order  perturbations.  For  rectangular  domains,  fifth 
order  perturbation  affords  no  increase  in  accuracy.  Though  the  accuracy  achieved  using  fourth  or  fifth 
order  perturbation  may  not  be  needed,  the  higher  order  perturbations  develop  confidence  in  the  conver¬ 
gence  of  the  predicted  eigensolutions  through  the  decreased  magnitude  of  the  higher  order  terms. 

Comparisons  of  perturbation  predictions  and  exact  eigenvalues  are  shown  in  Table  3  for  rectangular 
domains.  For  square  domains,  all  predicted  values  differ  from  the  exact  values  by  less  than  1%,  and  the 
agreement  is  also  excellent  for  ^  =  0.95  and  ^  =  0.9,  The  lowest  four  eigenvalues  provide  excellent  esti¬ 
mates  for  ^  =  0.7, 0.8,  as  shown  in  Fig.  5.  Where  the  predicted  and  exact  values  differed  substantially,  the 
failure  of  the  higher  order  perturbations  to  approach  zero  in  the  asymptotic  expansion  is  evident. 

It  is  interesting  that  two  distinct  circular  domain  eigenvalues  can  merge  to  form  a  degenerate  eigen¬ 
value  pair  on  a  square  domain.  For  instance,  the  (  m ,  n )  =  (  1, 0)  circular  domain  eigenvalue  and  one  of 
the  degenerate  ( m ,  n )  =  (0, 2)  circular  domain  eigenvalues  merge  to  form  the  degenerate  eigenvalue  pair 
ila  =4.9673  in  the  square  domain.  In  contrast,  the  degenerate  eigenvalue  pairs  Q.a  =  3,5124  and 
Q.a  =  5.6636  in  the  square  domain  evolve  from  the  degenerate  eigenvalue  pairs  (m,n)  =  (0, 1)  and 
( m , « )  =  ( 0, 3)  in  the  circular  domain.  Perturbation  predicts  the  splitting  of  the  degenerate  square 
domain  eigenvalues  (Fig.  5). 

Conclusions 

1.  Eigensolutions  of  the  wave  equation  with  perturbations  of  the  boundary  conditions  are  derived  by  exact 
solution  of  the  sequence  of  perturbation  problems  through  fifth  order.  Perturbations  of  the  domain  from 
circular  and  variation  of  boundary  condition  parameters  along  the  boundary  curves  are  included  in  the  class 
of  perturbations  for  which  the  method  applies.  Exactness  of  the  perturbation  solutions  means  no  approxi¬ 
mation  is  introduced  other  than  truncation  of  the  asymptotic  series  (9,  10), 


1496 


2.  The  derived  solution  offers  a  combination  of  analytical  and  computational  advantages: 

•  exact  perturbation  through  fifth  order  yields  excellent  accuracy  for  perturbations  of  substantial  magni¬ 
tude  (such  as  the  elliptic^  and  rectangular  domain  perturbation  examples); 

•  Fourier  representation  of  the  perturbations  allows  treatment  of  general  continuous  or  discontinuous 
asymmetries; 

•  algebraic  simplicity  of  the  results  permits  convenient  use  of  the  eigensolutions  in  applications  such  as 
inverse  and  forced  response  problems; 

•  admissible  functions  as  required  in  Ritz-Galerkin  analysis  are  not  needed; 

•  results  are  easily  derived  and  verified  using  computer  algebra  software. 

3.  Rules  governing  splitting  of  the  degenerate  unperturbed  eigenvdues  are  derived  at  both  first  and  second 

orders  of  perturbation.  These  rules  take  simple  algebraic  fonns  in  terms  of  the  Fourier  coefficients  of  a 

general  asymmetry.  The  rule  for  first  order  eigenvalue  splitting  is  such  that  it  can  be  applied  by  inspection. 


1  J.  R.  Kuttler  and  V.  G.  Sigillito,  “Eigenvalues  of  the  Laplacian  in  Two  Dimensions,”  SIAM  Review, 
vol.  26,  pp.  163-193,  1984. 

2.  J.  Mazumdar,  *  ‘A  Review  of  Approximate  Methods  for  Determining  the  Vibrational  Modes  of  Mem¬ 
branes,”  Shock  and  Vibration  Digest,  vol.  7,  pp.  75-88,  1975. 

3.  J.  Mazumdar,  ‘  ‘A  Review  of  Approximate  Methods  for  Determining  the  Vibrational  Modes  of  Mem¬ 
branes,”  Shock  and  Vibration  Digest,  vol.  1 1,  pp.  25-29, 1979. 

4.  J.  Mazumdar,  ‘  ‘A  Review  of  Approximate  Methods  for  Determining  the  Vibrational  Modes  of  Mem¬ 
branes,”  Shock  and  Vibration  Digest,  vol.  14,  pp.  11-17, 1982. 

5.  S.  B.  Roberts,  “The  Eigenvalue  Problem  for  Two-Dimensional  Regions  with  Irregular  Boundaries,” 
Journal  of  Applied  Mechanics,  vol.  34,  no.  3,  pp.  618-622,  1967. 

6.  D.  D.  Joseph,  “Parameter  and  Domain  Dependence  of  Eigenvalues  of  Elliptic  Partial  Differential 
Equations,”  Arch.  Rat.  Meek.  Anal.,  vol.  24,  pp.  325-351, 1967. 

7.  J.  Mazumdar,  “Transverse  Vibration  of  Membranes  of  Arbitrary  Shape  by  the  Method  of  Constant- 
Deflection  Contours,”  Journal  of  Sound  and  Vibration,  vol.  27,  pp.  47-57,  1973. 

8.  P.  M.  Morse  and  H.  Feshbach,  Methods  of  Theoretical  Physics,  McGraw-Hill,  1953. 

9.  A.  H.  Nayfeh,  Introduction  to  Perturbation  Techniques,  pp.  426-431,  J.  Wiley  &  Sons,  1981. 

10.  R.  G.  Parker  and  C.  D.  Mote,  Jr.,  “Exact  Boundary  Condition  Perturbation  Solutions  in  Eigenvalue 
Problems,”  Journal  of  Applied  Mechanics,  vol.  63,  pp.  128-135,  March  1996. 

11.  R.  G.  Parker  and  C.  D.  Mote,  Jr.,  “Exact  Higher-order  Boundary  Condition  Perturbation  in  Eigen¬ 
value  Problems,”  Journal  of  Applied  Mechanics,  in  preparation  1996. 

12.  R.  G.  Parker  and  C.  D.  Mote,  Jr.,  “Exact  Perturbation  for  the  Vibration  of  Almost  Annular  or  Circu¬ 
lar  Plates,”  Journal  of  Vibration  and  Acoustics,  vol.  118,  pp.  436-445,  July  1996. 

13.  S.  D.  Daymond,  ‘  ‘The  Principal  Frequencies  of  Vibrating  Systems  with  Elliptic  Boundaries,’  ’  Quart. 
Joum.  Mech.  and  Applied  Math,  vol.  VIII,  pp.  361-372,  1955. 

14.  B.  A.  Troesch  and  H.  R.  Troesch,  “Eigenfirequencies  of  an  Elliptic  Membrane,”  Mathematics  of 
Computation,  vol.  27,  pp.  755-765, 1973. 


II 

O 

e  =  0.5 

e=0.6 

e=0.7 

II 

o 

bo 

ns 

It 

O 

VO 

e  =0.9611 

Exact 

2.5165 

2.5968 

2.7202 

2.9215 

3.2933 

4.2151 

6.2432 

Pert.  (5) 

2.5165 

2.5968 

2.7202 

2.9215 

3.2936 

4.2287 

7.2790 

%  Error 

0.00% 

0.00% 

0.00% 

0.00% 

0.01% 

0.32% 

17.% 

(Doia)i 

Pert.  (4) 

3.9212 

3.9864 

4.0878 

4.2559 

4.5726 

5.3318 

(f2oi^)2 

Pert.  (4) 

4.0956 

4.2822 

4.5646 

5.0154 

5.8259 

7.7601 

Table  1:  Comparison  of  the  fundamental  elliptical  domain  eigenvaluecomputed  using  pertur¬ 
bation  to  the  exact  solution  of  Daymond  [13].  Perturbed  eigenvalues  evolving  from  the  0 
nodal  circle,  1  nodal  diameter  circular  domain  eigenvalues  are  also  presented,  e  denotes  the 
eccentricity  of  the  ellipse.  Numbers  in  parentheses  indicate  the  order  of  perturbation. 


1497 


^=1 

^=0.9 

^=0.8 

^=0.7 

4=0.6 

4=0.5 

4=0.4 

4=0.3 

Exact 

2.2214 

2.3481 

2.5145 

2.7391 

3.0531 

3.5124 

4.2295 

5,4665 

Pert.  (5) 

%  Error 

2.5183 

0.15%  . 

2.7460 

0.25% 

3.0665 

0.44% 

3.5597 

1.3% 

Pert.  (4) 

%  Error 

2.5109 

-0.14% 

2.7325 

-0.24% 

3.4797 

-0.93% 

4.1413 

-2.1% 

4.9888 

-8.7% 

Pert.  (3) 

%  Error 

2.2127 

-0.40% 

2.3394 

-0.37% 

3.4672 

-1.3% 

4.0995 

-3.1% 

Pert.  (2) 

%  Error 

2.2479 

1.2% 

2.3753 

1.2% 

n 

3.0923 

1.3% 

5.9903 

9.6% 

Pert.  (1) 

%  Error 

2.1430 

-3.5% 

2.2608 

-3.7% 

2.4049 

-4.4% 

2.5859 

-5.6% 

2.8209 

-7.6% 

3.1400 

-11.% 

3.6013 

-15.%- 

4.3353 

-21.% 

_ 

Table  2:  Comparison  of  the  fundamental  rectangular  domain  eigenvalue  computed  using 
perturbation  to  exact  values.  ^  =  b/a  denotes  the  aspect  ratio  of  the  rectangle.  The  numbers 
in  parentheses  indicate  the  order  of  perturbation. 


(m,  n) 

(0,0) 

(0,  l)i 

(0.  1)2 

(0,2)2 

(0,2)i 

(1,0) 

(0,3)i 

(0,  3)2 

Exact 

2.2214 

3.5124 

3.5124 

4.4429 

4.9673 

4.9673 

5.6636 

5.6636 

^=1 

Pert. 

2.2243 

3.5081 

3.5123 

4.4322 

4.9679 

4.9656 

5.7064 

5.7071 

%  Err. 

0.13% 

-0.12% 

0.00% 

-0.24% 

0.01% 

-0.03% 

0.76% 

0.76% 

Exact 

2.2806 

3.5502 

3.6610 

4.5613 

4.9941 

5.2032 

5.7569 

5.8716 

4  =  0.95 

Pert. 

2.2836 

3.5442 

3.6553 

4.5499 

5.2027 

5.8693 

5.8873 

%ErT. 

0.13% 

-0.17% 

-0.16% 

-0.25% 

0.00% 

2.0% 

0.27% 

Exact 

2.3481 

3.5939 

3.8278 

4.6962 

5.0252 

5.4665 

5.8644 

6.1062 

4  =  0.9 

Pert. 

2.3511 

3.5879 

3.8218 

4.6831 

4.9148 

5.5454 

6.1071 

6.1390 

%  Err. 

0.13% 

-0.17% 

-0.16% 

-0.28% 

-2.2% 

1.4% 

4.1% 

0.54% 

Exact 

2.5145 

3.7047 

4.2295 

5.0290 

6.0963 

6.1342 

4  =  0.8 

Pert. 

2.5183 

3.6984 

4.2219 

5.0063 

3.9446 

6.6318 

6.7169 

%  Err. 

0.15% 

-0.17% 

-0.17% 

-0.45% 

-1.9% 

-35.% 

8.1% 

0.61% 

Exact 

2.7391 

3.8607 

4.7549 

5.4783 

5.2194 

6.9128 

6.5076 

7.4289 

4=0.7 

Pert. 

2.7460 

3.8493 

4.7478 

5.4315 

8.7571 

imag. 

6.1737 

7.0269 

%  Err. 

0.25% 

-0.29% 

-0.85% 

68.% 

-5.1% 

-5.4% 

Table  3:  Comparison  of  rectangular  domain  eigenvalues  from  perturbation  to  exact  values.  ^  =  b/a 
is  the  aspect  ratio,  m  and  n  are  the  numbers  of  nodal  circles  and  nodal  diameters  in  the  circular 
domain  eigenfunction  from  which  the  corresponding  rectangular  domain  eigenfunction  evolves. 


1498 


Q  a 

nnn 


Figure  3:  Elliptical  domain  eigenvalues.  The  subscript  mn  denotes  the  number  of  nodal  circles  (m) 
and  nodal  diameters  (n)  in  the  circular  domain  eigenvalue,  a  is  the  semi-major  axis  of  the  ellipse.  The 
solid  lines  are  from  the  exact  solutions  of  Daymond  [13]  and  Troesch  and  Troesch  [14].  The  symbols 
are  values  predicted  by  perturbation. 


1499 


Figure  4:  Rectangle  of  aspect  ratio  ^  =  0.9.  The 
approximate  rectangle  is  a  10  term  Fourier  approx¬ 
imation.  The  circle  is  the  domain  for  perturbation. 


Figure  5:  Eigenvalues  of  a  rectangle  with  aspect  ratio  ^  =  b/a.  Solid  curves  denote  exact 
values.  Symbols  denote  values  predicted  by  perturbation. 


1500 


SUBSTRUCTURING  FOR  SYMMETRIC  SYSTEMS 

A.V.  Pesterev 


State  Institute  of  Physics  and  Technology 
13/7,  Prechistenka  St.,  Moscow,  119034  Russia 
E-mail:  sash.a@pesterev.msk.ru 


ABSTRACT 

It  is  known  that  the  problem  of  finding  the  spectrum  of  a  complex 
conservative  system  with  interaction  of  finite  rank  between  its  subsystems 
can  be  reduced  to  investigation  of  some  symmetric  characteristic  matrix. 
If  the  system  exhibits  symmetries  of  one  kind  or  another,  the  problem  can 
be  further  simplified.  It  is  shown  how  one  can  decompose  the  problem 
for  a  symmetric  system  by  using  results  of  the  representation  theory  of 
groups.  Considering  a  group  of  rigid  symmetries  of  the  system  and  its 
representation  in  the  interaction  space,  one  can  obtain  the  spectrum  of  the 
system  by  investigating  a  number  of  reduced  characteristic  matrices,  the 
sum  of  their  orders  being  equal  to  the  rank  of  the  interaction. 

1  INTRODUCTION 

It  is  well  known  that  the  problem  of  finding  dynamic  characteristics 
of  a  complex  structure  consisting  of  distributed  and/or  finite- dimensional 
subsystems  which  interact  at  a  finite  number  of  points  can  be  reduced  to 
investigation  of  some  matrix  of  relatively  small  order  (see,  e.g.,  [1-10]  and 
references  therein),  which  is  sometimes  referred  to  as  the  characteristic 
matrix.  If  the  system  exhibits  symmetries  of  one  kind  or  another,  the 
problem  can  be  simplified  from  the  computational  standpoint.  The  aim 
of  this  paper  is  to  develop  a  technique  for  investigation  of  such  systems. 
To  facilitate  discussions,  we  shall  restrict  our  consideration  to  conservative 
systems.  To  reduce  the  original  spectral  problem  to  the  problem  of  inves¬ 
tigation  of  a  characteristic  matrix,  we  will  apply  the  technique  used  in  the 
structural  analysis  method  [5,  6]. 


1501 


2  FORMULATION  OF  THE  PROBLEM 


Let  us  consider  a  linear  complex  mechanical  system  consisting  of  a  finite 
number  of  distributed  and/or  finite-dimensional  subsystems  interacting  at 
a  finite  number  of  points.  Let  us  write  the  equation  governing  small  sta¬ 
tionary  vibrations  of  jth  isolated  subsystem  in  the  following  operator  form 

Aj{X)Xj  =  Fj,  (1) 

where  Fj  is  a  vector  function  of  amplitudes  of  external  harmonic  general¬ 
ized  forces  acting  on  the  subsystem,  Xj  is  an  amplitude  response  vector 
function,  A  =  w  is  a  circular  frequency,  Aj(A)  is  a  self-adjoint  operator. 
In  the  case  of  a  distributed  subsystem,  Aj(A)  is  a  differential  operator;  for 
a  finite- dimensional  subsystem,  it  is  given  by  a  matrix. 

Let  the  subsystems  interact  elastically  at  a  finite  number- of  points 
(e.g.,  they  are  connected  to  each  other  by  means  of  a  finite  number  of 
conservative  springs).  Let  us  denote  by  X  the  i?-component  vector  func¬ 
tion  of  responses  of  the  system,  the  jth  component  of  X  being  vector 
function  Xj.  Clearly,  the  interaction  forces  depend  on  displacements  of 
only  those  points  that  take  part  in  the  interaction;  i.e.,  they  depend  on 
some  finite-dimensional  vector  Y  e  rather  than  on  response  vector 
function  X,  which,  in  the  general  case,  belongs  to  some  functional  space. 
Denote  by  S  an  operator  transforming  X  into  the  vector  Y:  Y  =  SX.  Let 
the  interaction  be  given  by  a  stiffness  matrix  K  such  that  the  amplitude 
vector  of  generalized  interaction  forces  F  C  R^  is  given  by  the  equation: 
F  =  —KY.  As  the  interaction  is  assumed  to  be  conservative,  the  matrix  K 
is  positive  semidefinite.  To  transform  vector  F  into  the  right-hand  sides 
of  equations  (1),  the  adjoint  operator  S*  is  used  (see  [5,  6]  for  more  de¬ 
tail).  Assuming  zero  external  forces,  one  arrives  at  the  following  equation 
governing  free  stationary  vibration  of  the  system  under  consideration 

(A(A)  +  S*KS)X  =  0  ,  (2) 

where  A(A)  is  a  diagonal  operator  matrix  A(A)  =  diag[Aj(A)<5jjfe]j^j;._i  gov¬ 
erning  vibration  of  the  aggregate  of  the  non-interacting  subsystems.  The 
discrete  spectrum  of  the  system  is  defined  to  be  the  set  of  numbers  A  such 
that  equation  (2)  has  non-trivial  solutions.  More  detailed  discussions  con¬ 
cerning  such  a  formulation  of  spectral  problems,  as  well  as  some  examples, 
are  given  in  [5,  6]. 

It  is  well  known  (e.g.,  [1,  3-6])  that  if  operator  A“^(A)  can  be  calcu¬ 
lated,  the  problem  of  finding  the  spectrum  can  be  reduced  to  the  problem 
of  investigating  some  characteristic  matrix.  The  matrices  can  be  defined 
differently,  but  all  of  them  have  an  important  property:  if  A~^(A)  exists, 
problem  (2)  has  non-trivial  solutions  if,  and  only  if,  the  characteristic  ma¬ 
trix  is  singular. 


1502 


In  this  paper,  we  will  take  advantage  of  the  definition  of  the  charac¬ 
teristic  matrix  used  in  the  structural  analysis  method  [5-7].  As  the  stiff¬ 
ness  matrix  K  is  positive  semidefinite,  it  ^n  be  factored  in  the  form  [11]: 
K  =  a,  where  a  is  Ny.N  matrix,  and  N  is  the  rank  of  matrix  K.  The 
characteristic  matrix  of  order  N  is  then  defined  by  the  equation 

Q(X)^  I  +  aSA-\\)S*a^  .  (3) 

Properties  of  eigenvalues  and  eigenvectors  of  the  matrix  that  guarantee 
finding  all  eigenfrequencies  of  the  system  in  a  given  frequency  range  are 
discussed  in  [6].  To  save  room,  we  will  not  discuss  here  the  ways  of  in¬ 
vestigation  of  those  values  of  A  at  which  A"^(A)  does  not  exist  (detailed 
discussions  of  this  case  as  applied  to  nonsymmetric  structures  can  be  found 
in  [8,  9]),  as  this  has  no  effect  on  the  decomposition  discussed. 

Let  now  the  subsystems  (or  part  of  them)  be  identical  or  can  be  divided 
into  a  few  groups  of  identical  ones  and  interact  in  such  a  way  that  the  com¬ 
plex  system  exhibits  symmetries  of  one  kind  or  another.  It  is  reasonable 
to  suggest  that  these  symmetries  can  be  used  to  simplify  the  problem  and 
also  understand  and  classify  solutions.  The  technique  to  be  discussed  in 
this  paper  allows  one  to  decompose  matrix  (3)  into  a  number  of  matrices 
of  lesser  orders  such  that  the  spectrum  of  the  system  can  be  obtained  by 
investigating  these  matrices  separately. 

3  DECOMPOSITION  OF  THE  INTERACTION 

SPACE 

Let  Cj  be  a  group  of  symmetry  of  the  immovable  system,  elements  of 
the  group  g  ^  G  being  either  rotations  or  reflections  that  transform  the 
system  into  itself.  Clearly,  as  we  consider  systems  with  a  finite  number  of 
interaction  points,  G  can  be  only  a  finite  group.  For  the  same  reason,  we 
can  consider  a  representation  of  the  group  in  a  finite-dimensional  space. 
Actually,  it  follows  from  the  results  discussed  in  [5,  6]  that  at  a  given  eigen- 
frequency  c<;o  J^ere  is  one-to-one  relationship  between  eigenfunctions  of  the 
system  and  A/’-dimensioi^l  vectors  from  the  kernel  of  the  matrix  Q{Gl) 
KerQ(a;Q).  However,  as  A/’-dimensional  space,  come  into  existence  as  a  re¬ 
sult  of  a  formal  procedure  of  the  factorization  of  the  stiffness  matrix  A,  it 
may  be  difficult  to  build  a  representation  of  the  group  in  this  space.  There 
are  two  physical  A-dimensional  spaces  relevant  to  the  problem,  namely, 
the  space  of  displacements  of  the  points  at  which  the  subsystems  interact 
with  each  other  and  the  space  of  generalized  interaction  forces.  Either  of 
them  can  be  used  to  build  a  representation  of  the  group.  Let,  for  definite¬ 
ness,  it  be  the  space  of  displacements  Y  =  SX\  we  denote  it  by  letter  L 
and  will  call  the  “interaction”  space.  It  is  evident  from  general  consider¬ 
ations  that  at  a  given  eigenfrequency  there  exist  one-to-one  relationship 


1503 


between  the  eigenfunctions  of  the  system  and  vectors  of  the  corresponding 
displacements  of  the  connection  points. 

Let  the  system  free  vibrate  at  some  eigenfrequency,  and  let  y  6  L  be 
the  corresponding  vector  of  the  displacements  of  the  connection  points. 
From  this  point  on,  we  will  call  it,  for  brevity,  the  “eigenvector”  of  the  sys¬ 
tem.  Any  rotation  or  reflection  g  e  G  transforms  the  eigenvibration  into 
some,  generally  speaking,  different  eigenvibration,  the  eigenvector  Y  being 
transformed  into  some  other  eigenvector  Let  us  introduce  linear  oper¬ 
ators  (matrices)  T{g)  by  the  equation:  T{g)Y  =  g  Clearly,  T(g) 
is  a  real  unitary  representation  of  group  G  and  can  be  constructed  for  any 
particular  configuration  of  the  interaction  points  and  a  given  coordinate 
system. 

As  will  be  seen  from  the  next  section,  the  decomposition  of  the  spectral 
problem  can  be  obtained  if  we  succeed  in  decomposing  the  space  L  into  a 
number  of  mutually  orthogonal  real  subspaces 

L  =  Li  -f  •  ■  •  +  Ljn  (4) 

such  that  any  eigenvector  of  the  system  Y  could  be  represented  in  the  form 
y  —  where  Yj  e  Lj  is  also  an  eigenvector  of  the  system  corresponding 

to  the  same  eigenfrequency.  In  other  words,  we  need  such  a  decomposition 
that  either  an  eigenvector  Y  belongs  to  a  subspace  Lj  or  (in  the  case  of 
a  multiple  eigenfrequency)  there  exist  such  elements  of  the  group  gk  and 
real  numbers  (Sk^  which  are  not  all  zero,  that  Ylkl^kT{gk)Y  G  Lj.  Because 
of  lack  of  room,  we  restrict  our  discussions  to  substantiating  reasoning  and 
give  the  results  without  proofs. 

Let  Ti{g), . . . ,  Tq[g)  be  real  nonequivalent  irreducible  unitary  represen¬ 
tations  of  the  group  G  and  representation  Tj{g)  of  dimension  Vj  occurs  rrij 
times  in  T{g).  The  following  lemma  is  valid. 

Lemma  1  For  any  gwen  j ,  there  exist  such  g^  G  G  and  real  numbers  13k, 
/c  =  1, . . . ,  Tj  +  1,  that  Ylk  KTj{gk)  =  0. 

The  proof  is  based  upon  application  of  Cayley-Hamilton  theorem  [11]  to 
the  matrix  Tj{g)  with  g  being  an  element  of  a  cyclic  subgroup.  It  is  well 
known  [12]  that  the  space  L  can  be  decomposed  into  mutually 

orthogonal  invariant  subspaces  each  of  which  is  transformed  by  one  of 
the  irreducible  representations.  Let  us  denote  by  Lj  {j  =  1,...,^)  the 
direct  sum  of  all  invariant  subspaces  transformed  by  the  representations 
equivalent  to  the  real  representation  Tj{g).  Using  Lemma  1,  one  can  prove 
the  following  theorem. 

Theorem  1  For  any  eigenfrequency,  an  eigenvector  Y  either  belongs  to 
one  of  the  suhspaces  Lj  or  can  be  represented  in  the  form  Y  =  YlYj,  where 
Yj  G  Lj  is  also  an  eigenvector  of  the  system.  If  an  eigenvector  of  the 
system  belongs  to  a  suhspace  Lj,  then  the  multiplicity  of  the  eigenfrequency 
is  equal  to  rj. 


1504 


Thus,  we  have  obtained  the  desired  decomposition  (4)  of  space  L  with  m 
being  equal  to  the  number  of  real  nonequivalent  irreducible  representations 
of  point  group  G.  To  take  advantage  of  decomposition  (4),  one  needs 
projectors  Pj  onto  the  subspaces  Lj  :  PjL  =  Lj.  The  projector  on  the 
subspace  Lj  is  given  by  the  formula  [12,  Chapter  4] 

=  (5) 

g^G 

where  n  is  the  order  of  the  group  G,  and  Xi(5')  is  the  character  of  Tj{g). 

It  turns  out  that  under  certain  conditions  some  of  subspaces  Lj,  which 
are  transformed  by  representations  whose  dimensions  are  greater  than  one, 
can  be  decomposed  in  turn. 

Theorem  2  Let  Tj{g)  be  a  real  irreducible  representation  of  a  group  G  of 
dimension  rj  >  I,  and  let  there  exist  an  element  of  the  group  go  of  order 
T-j  such  that  the  matrix  Tj{go)  has  all  different  complex  eigenvalues.  Then 
the  subspace  Lj  can  he  decomposed  into  Pj  subspaces,  where,  for  even  r^, 
Pj  =  rff2  +  1,  and,  for  odd  rj,  pj  =  {rj  +  l)/2. 

The  theorem  allows  one  to  obtain  more  subtle  decomposition  (4)  with  m 
being  greater  than  the  number  of  real  irreducible  representations. 

Let  us  obtain  formulas  for  projectors  onto  the  subspaces  of  a  decompos¬ 
able  subspace  Lj.  As  mentioned  above,  Lj  is  the  direct  sum  of  subspaces 
L),  k  =  l,...,mj,  transformed  by  the  representations  equivalent  to  Tj{g). 
Let  us  consider  for  a  while  Lj  as  complex  subspaces.  It  is  evident  that 
eigenvalues  of  the  matrix  Tj{go)  are  numbers  (*,  i  =  0, 1, . . .  ,rj  -  1,  where 
C  is  a  primitive  rf  root  of  1  =  1).  Let  e\  be  the  corresponding  eigen¬ 

vectors  of  matrix  Tj{go)  belonging  to  the  subspace  L^.  If  rj  >  2,  some 
vectors  ef  are  complex,  and,  for  each  complex  vector  ef ,  there  is  its  com¬ 
plex  conjugate  vector  ef.  Denote  by  ^j  •  (z  =  0, 1, . . . ,  Pj  —  i)  the  subspace 
of  Lj  spanned  either  by  vector  ef,  if  is  real,  or  by  two  vectors  ef  and 
ef.  Denote  by  Ljj  the  direct  sum  of  subspaces  /j^^,  k  =  l,...,mj.  It  is 
the  decomposition  of  Lj  into  pj  real  subspaces  Lj^i  that  is  established  in 
Theorem  2. 

Let  us  take  the  above  eigenvectors  e-’’,  i  =  0,l,...,rj  —  1  for  the  or¬ 
thonormal  basis  in  subspace  Lj  and  write  the  representation  Tj{g)  in  ma¬ 
trix  form  Tffg)  =  z,  /  =  0, 1, . . . ,  rj  -  1,  which  is  the  same  in  all 

subspaces  Lj,  fc  =  1, . . .  ,mj.  Using  the  results  of  [12,  Chapter  4],  one  can 
obtain  formulas  for  projectors  Pj^i  on  the  subspaces  Lj^i 

Pj-  =  i'MTig)  ,  Z  =  0  and  z  =  p,  -  1  (for  even  rj)  ,  (6) 

g^G 

Pj.i  =  -  Z(«y<7)  +  iU9))T{9)  .otherwise.  (7) 

^  geG 


1505 


Note  that  the  eigenvectors  of  the  system  Yj,i  6  Lj^i  exist  or  do  not  exist 
in  all  subspaces  Lj^i  simultaneously,  so  that  the  multiplicity  of  the  corre¬ 
sponding  eigenfrequency  of  the  system  is  equal  to  rj  as  before. 

4  DECOMPOSITION  OF  SPECTRAL  PROBLEM 

In  the  previous  section,  we  have  decomposed  space  L  of  displacements 
of  the  connection  points.  One  could  take  for  L  the  space  of  generalized 
interaction  forces  and  obtain  the  same  decomposition.  It  is  not  difficult  to 
prove  that  if  the  system  free  vibrates  and  the  corresponding  eigenvector 
Y  belongs  to  a  subspace  Lj  then  the  corresponding  vector  of  interaction 
forces  F  =  —KY  also  belongs  to  the  same  subspace  Lj.  This  allows  one 
to  introduce  matrices  Kj  =  PjKPj.,  ^’  =  1, . . . ,  m,  and  consider  m  spectral 
problems  of  the  form  (2)  in  which  the  matrix  K  is  replaced  by  Kj]  here 
projector  Pj  is  given  either  by  formula  (5)  or  by  (6)-(7).  Matrices  Kj 
can  also  be  presented  in  the  factorized  form:  Kj  =  aja,  where  aj  is 
NjxN  matrix,  and  Nj  is  the  rank  of  Kj.  Let  us  define,  analogously  to  (3), 
matrices  Qj{X)  of  orders  Nj  by  the  formula 

Qj(X)  =  !  +  aiSA-^X)S‘aJ  .  (8) 

It  turns  out  that  to  obtain  the  spectrum  of  the  system,  one  can  investigate 
separately  matrices  Qj(A)  instead  of  one  matrix  Q{X).  Let  Uj  be  the  set 
of  values  of  A  such  that  matrix  (5i(A)  exists  and  is  singular.  Denote  by  U 
the  union  of  the  sets  f/j,  ;  =  1, . . . ,  m.  If  some  A  occurs  r  times  in  C/,  we 
say  that  it  has  multiplicity  r.  __ 

Theorem  3  1.  The  sum  of  orders  Nj  ofQj{X)  is  equal  to  the  order  N  of 

QW-  ' 

2.  The  set  of  the  eigenvalues  of  the  system  different  from  those  of  the 
isolated  subsystems  coincide  with  the  set  U . 

The  eigenvalues  of  the  isolated  subsystems  are  investigated  separately.  The 
approach  used  in  [5,  6,  8,  9]  for  nonsymmetric  systems  can  be  modified  and 
applied  to  symmetric  ones. 

Remark,  In  the  general  Cctse  of  a  3D  structure  consisting  of  several 
substructures,  the  subsystem  points  that  take  part  in  the  interaction  can 
be  divided  into  a  few  independent  groups,  called  “interaction  sections,” 
such  that  the  interaction  forces  acting  on  the  subsystems  at  the  points 
belonging  to  a  certain  interaction  section  depend  on  displacements  of  only 
those  points  that  belong  to  this  interaction  section.  This  subdivision  of 
the  interaction  points  into  independent  interaction  sections  implies  a  pri¬ 
ori  decomposition  (not  connected  with  the  symmetry  of  the  system)  of  the 
interaction  space  and  results  in  a  block-diagonal  form  of  the  stiffness  ma¬ 
trix  K  (under  an  appropriate  numbering  of  the  interaction  points).  This 


1506 


implies  also  that  we  can  build  representations  of  the  group  in  different 
interaction  sections  separately. 


5  PROJECTORS  FOR  THE  GROUPS  AND 


As  an  illustration,  let  us  decompose  the  interaction  space  and  calcu¬ 
late  projectors  for  two  cases  of  symmetry.  First,  consider  a  system  having 
n-fold  axis  of  symmetry  such  that  rotations  about  the  axis  through  an¬ 
gles  A:  =  0, 1, . . .  ,n  -  1,  transform  the  system  into  itself.  The 

rotations  form  the  cyclic  abelian  group  On  of  order  n.  It  has  [12]  n 

one-dimensional  complex  irreducible  representations.  Combining  pairs  of 
complex  conjugate  representations  into  real  two-dimensional  representa¬ 
tions,  one  obtains  m  real  irreducible  representations,  where,  for  even  n, 
m  —  n/2  -f  1,  two  representations  being  one- dimensional  and  the  others 
being  two-dimensional;  for  odd  n,  m  =  (n  l)/2,  with  one  representation 
being  one-dimensional.  Calculating  the  characters  of  the  representations 
and  applying  formula  (5),  one  obtains  formulas  for  the  projectors  onto  the 
subspaces  Lq  and  Lnf2  (for  even  n)  corresponding  to  the  one- dimensional 
real  representations 

P,  =  iy:T(Ci')  .  Pj  =  iE(-l)‘r{0  (even  n)  ,  (9) 

and  onto  the  subspaces  Lj  corresponding  to  the  two-dimensional  real  rep¬ 
resentations 


2  ^27rkj 

Pj  =  -  )r(cj 

k=0  ^ 


,  J  = 


1. .  . . ,  I  —  1,  for  even  n, 

1. .  .  . ,  for  odd  n. 


(10) 


As  a  second  example,  let  us  consider  the  group  of  symmetry  Cnv,  which 
also  describes  symmetry  arrangements  with  one  n-fold  axis.  But  apart 
from  the  rotations,  there  are  n  reflections  transforming  the  system  into 
itself.  The  order  of  the  group  is  equal  to  2n.  Let  us  choose  the  elements 
of  the  group  as  follows:  Cj  =  e,  . . . ,  ayCl^ . . .  where 

CTy  is  a  reflection  in  a  certain  symmetry  plane  passing  through  the  n-fold 
symmetry  axis.  It  is  known  [12]  that  for  odd  n  this  group  has  two  one¬ 
dimensional  (To,i  and  To, 2)  and  (n  -  l)/2  two-dimensional  real  nonequiv¬ 
alent  irreducible  representations.  For  even  n,  it  has  four  one-dimensional 
(To,i,  To,2,  Ti,!,  and  Tb,2)  and  (n  -  2)/2  two-dimensional  representations. 
Calculating  the  characters  of  the  one-dimensional  representations  and  ap¬ 
plying  formula  (5),  one  obtains  the  projectors  onto  the  subspaces  corre¬ 
sponding  to  the  one-dimensional  representations 


YE(T(C‘)  +  T(a.O) 

k=0 


(11) 


1507 


(12) 


^o,2  =  ^i:(r(c*)-r(c7„0), 

k=0 

and  (if  n  is  even) 

=  ;r  £(-1)'=  (net)  +  T(<r„C'i))  ,  (13) 

fc=0 

Ph^  =  r  £(-1)''  {net)  -  T(<r„C„‘))  .  (14) 

k=0 

It  is  not  difficult  to  show  that  bases  in  the  spaces  Lj  corresponding 
to  the  two-dimensional  representations  Tj{g)  can  be  chosen  in  such  a  way 
that  the  matrices  Tj{cry)  are  diagonal:  Tj{cry)  =  diag[l,— lj.  This  means 
that  the  two-dimensional  representations  Tj{g)  satisfy  the  conditions  of 
Theorem  2  with  go  =  <7^,  and,  hence,  each  of  the  subspaces  Lj  can  be 
decomposed  into  the  sum  of  two  real  subspaces  Lj  =  Lj^  +  Lj^i .  Applying 
formula  (6),  one  obtains  the  projectors  onto  the  subspaces  Lj^o  and  Lj^i  : 

P,.0  =  i  E  cosl"^)  {T{Ct)  +  n<^.Ct))  ,  (15) 

^  A:=0  ^  ^ 


where 


Pi,i  =  -  E  cos(?^)  (T(C,t)  -  na„Ct))  . 

^  k=0  ^ 


J 


1, . . . ,  for  even  n, 
1, . . . ,  for  odd  n. 


(16) 


6  AN  EXAMPLE  OF  CONSTRUCTION  OF  A 
REPRESENTATION 

In  the  previous  section,  we  obtained  formulas  (9)-(16)  for  the  projec¬ 
tors  for  the  symmetry  groups  Cn  and  Cnv  To  take  advantage  of  these 
formulas  for  a  given  symmetric  structure,  it  is  required  only  to  construct 
a  representation  of  the  group  in  the  corresponding  interaction  space.  The 
aim  of  this  section  is  to  demonstrate,  by  way  of  a  simple  example,  how 
such  a  representation  can  be  constructed. 

Let  a  symmetric  structure  consist  of  seven  interacting  subsystems,  and 
let  its  symmetry  be  described  by  the  group  Cey.  Let  the  6-fold  axis  of 
symmetry  be  perpendicular  to  the  plane  of  the  sheet.  Keeping  in  mind 
Remark  in  Section  4,  let  us  consider  one  interaction  section  of  the  sys¬ 
tem.  Let  the  interaction  points  belonging  to  this  section  be  located  at 
the  vertices  of  a  regular  hexagon  as  shown  in  Fig.  1.  Here  the  circles 
denote  the  interaction  (connection)  points  of  the  subsystems  belonging  to 


1508 


yi 


1 


Z2 


6 

o 


o 


2 


i 

7 


O 

5 


4 

O 


Fig.  1:  An  interaction  section  for  a  system  with  C^v  symmetry  group 

the  considered  interaction  section.  The  kind  of  the  subsystems  (finite-  or 
infinite-dimensional)  is  of  no  concern  at  this  stage,  since  a  representation  is 
constructed  in  a  hnite-dimensional  interaction  space.  Note,  however,  that 
the  configuration  shown  in  Fig.  1  could  represent,  e.g.,  a  section  of  a  bundle 
of  seven  identical  3D  beams  by  a  plane  perpendicular  to  the  longitudinal 
axes  of  the  beams. 

Let  the  points  be  connected  to  each  other  through  some  springs,  and 
the  interaction  be  described  by  a  stiffness  matrix  of  interaction  K.  To 
construct  a  representation,  a  particular  configuration  of  springs  is  of  no 
importance.  It  is  required  only  that  this  configuration  satisfy  the  same 
symmetry  conditions.  That  is,  the  system  of  springs  transforms  into  itself 
under  any  rotations  and  reflections  belonging  to  the  considered  group  Cqv 

To  simplify  the  consideration  and  to  save  room,  we  will  assume  first 
that  the  interaction  points  are  allowed  to  vibrate  only  in  the  plane  of  the 
figure,  and  the  interaction  forces  also  belong  to  this  plane.  Thus,  the 
interaction  space  L  in  this  example  is  14-dimensional.  The  construction 
of  the  representation  is  simplified  if  we  represent  the  displacement  of  each 
interaction  point  in  its  own  coordinate  system,  the  origin  of  which  is  placed 
at  the  equilibrium  state  of  the  point.  The  coordinate  systems  associated 
with  point  1  and  central  point  7  are  shown  in  the  figure.  The  coordinate 
system  of  point  (A:  =  2, . . . ,  6)  is  obtained  from  that  of  point  1  by  rotation 
about  the  symmetry  axis  x  (passing  through  the  central  point)  through  the 


1509 


angle  Cq  =  The  vector  of  displacements  of  the  interaction  points  is 
given  hyY  =  [j/i,  ,  y?,  zj]'^  e  where  y*  and  Zk  are  the  coordinates 

of  the  displacement  of  the  ^th  point  in  its  own  coordinate  system. 

As  discussed  in  the  previous  section,  the  group  Cev  consists  of  12  el¬ 
ements.  Let  cr„  be  the  reflection  in  the  symmetry  plane  passing  through 
the  6-fold  axis  x  and  points  1  and  4,  and  Cq  be  the  counterclockwise  ro¬ 
tation  about  the  6-fold  symmetry  axis  x  through  the  angle  Cq.  Let  the 
elements  of  the  group  be  chosen  in  the  same  way  a.s  in  Section  5:  Cq~ 
. . .  .cTyCi.  Since  T(yiy2)  =  'R{gi)T{g2)ygi,g2  ^  G 
[12],  we  need  to  construct  only  the  matrices  r(C'|),  =  1, . . . ,  5,  and  T{ay). 

It  can  be  checked  directly  by  examining  Fig.  1  that  the  representation 
T  is  given  by 


■  E  0  0  0  0  0  0  ■ 

0  0  0  0  0  i;  0 

0  0  0  0  0  0 

T{cry)  =  0  0  0  0  0  0 

0  0  0  0  0  0 

0  0  0  0  0  0 

0  0  0  0  0  0 


0  /  0  0  0  0  0 

0  0  /  0  ‘0  0  0 

0  0  0  /  0  0  0 

T{Cl)  =  0000/00 

0  0  0  0  0  /  0 

/  0  0  0  0  0  0 

0  0  0  0  0  0  Fi 


'oo/ooool  fooooo/o' 

000/000  1  0  0  0  0  0  0 
0000/00  0/00000 

TiCl)=  0  0  0  0  0  /  0  ,..-,T(C'|)=  0  0  /  0  0  0  0  , 

1  0  0  0  0  0  0  000/000 

0/00000  0000/00 

000000  r2j  [oooooor5_ 

where  0  stands  for  the  2x2  matrix  of  zeros,  I  is  the  2x2  identity  matrix, 

E  is  the  diagonal  matrix  E  =  diag[l,  —1],  and 


Clearly,  T(C'g)  is  the  identity  matrix  of  order  14,  and  the  other  five 
matrices  are  obtained  by  means  of  the  formula  —  T{cry)T{CQ). 

Now,  whatever  is  the  stiffness  matrix  corresponding  to  this  interaction 
section,  the  interaction  space  is  decomposed  into  eight  subspaces,  and  the 
projectors  onto  these  subspaces  are  easily  constructed  by  means  of  formulas 
(11)-(16)  at  n  =  6.  Repeating  this  procedure  for  the  other  interaction 
sections  and  applying  the  technique  discussed  in  Section  4,  we  obtain  eight 
characteristic  matrices  such  that  the  spectrum  of  the  original  problem  can 
be  obtained  by  investigating  these  matrices.  The  orders  of  these  matrices, 
in  the  general  case,  are  different  and  determined  by  a  particular  stiffness 


1510 


matrix  of  the  interaction.  However,  according  to  Theorem  3,  the  sum  of 
their  orders  is  equal  to  the  order  of  the  characteristic  matrix  of  the  system 
when  it  is  investigated  without  regard  for  its  symmetry. 

Note  finally,  without  going  into  detail,  that  the  consideration  of  a  more 
general  case,  the  case  of  space  displacements  of  the  interaction  points, 
presents  no  additional  problem.  Let,  for  example,  the  vibration  of  a  bundle 
of  seven  3D  beams  be  analyzed.  Let  the  beams  be  connected  to  each 
other  through  some  number  of  translational  and  torsional  springs  located 
in  a  plane  perpendicular  to  the  longitudinal  axes  of  the  beams.  Let  the 
interaction  section  of  the  bundle  be  as  shown  in  Fig.  1.  In  this  case,  the 
displacement  of  each  interaction  point  (which  is  actually  a  cross-section 
of  the  corresponding  beam)  is  described  by  six  (three  linear  and  three 
angular)  coordinates.  Introducing  the  third  axis  for  each  coordinate  system 
and  the  Eulerian  angles,  one  can  construct  the  representation  in  the  42- 
dimensional  interaction  space.  Writing  the  matrices  of  the  representation 
in  terms  of  6x6  matrix  blocks,  one  will  obtain  them  exactly  in  the  same 
form  as  in  the  previous  case  of  plane  vibrations.  Blocks  0  and  /,  in  this  case, 
are  6x6  zero  and  identity  matrices,  respectively;  E  is  a  diagonal  matrix 
of  ones  and  minus  ones;  and  Tk  describes  the  coordinate  transformation 
under  rotation  about  the  6-fold  symmetry  axis  through  the  angle  Cq.  A 
concrete  form  of  matrices  E  and  Tk  depend  on  the  chosen  enumeration  of 
the  six  coordinates  describing  the  position  of  the  point  in  the  interaction 
space. 

7  NUMERICAL  ILLUSTRATIVE  EXAMPLE 

The  aim  of  this  section  is  to  give  a  simple  numerical  example  of  decom¬ 
position  of  the  interaction  space.  Since  the  result  of  the  decomposition  of 
the  interaction  space  (i.e.,  the  number  of  nonempty  subspaces  and  their 
dimensions  and,  hence,  the  number  of  the  reduced  characteristic  matri¬ 
ces  and  their  orders)  does  not  depend  on  the  kind  and  complexity  of  the 
subsystems  and  is  determined  only  by  the  symmetry  group,  the  configu¬ 
ration  of  the  interaction  points,  and  the  stiffness  matrix  of  the  interaction 
between  the  subsystems,  we  will  consider  a  structure  consisting  of  the  sim¬ 
plest  subsystems — concentrated  masses. 

We  will  consider  a  system  possessing  the  symmetry  group  C&v  and  take 
advantage  of  the  representation  constructed  in  the  previous  section.  Let 
the  free-free  spring-mass  system  shown  in  Fig.  2  perform  plane  vibration. 
Here  the  circles  denote  identical  masses  m  =  1,  and  the  lines  connecting 
the  masses  denote  linear  springs  of  the  stiffness  k  =  1. 

The  direct  analysis  results  in  a  conventional  matrix  formulation  of  the 
spectral  problem.  The  eigenvalues  (eigenfrequencies  squared)  are  obtained 


1511 


1 

2 


3 


4 

Fig.  2:  A  spring-mcLSS  system  of  7  masses  and  12  springs 

by  means  of  solving  the  eigenvalue  problem 

{K-XM)h  =  0,  (17) 

where  M  is  the  identity  mass  matrix  of  order  14,  and  the  stiffness  matrix 
K  of  the  same  order  is  easily  constructed  by  means  of  Fig.  2.  The  direct 
computations  yield  3  rigid  body  and  11  elastic  modes.  The  eigenvalues 
corresponding  to  elastic  modes  are  0.849  (2),  1.000  (1),  1.229  (2),  2.000  (1), 
2.651  (2),  3.000  (1),  and  4.271  (2).  The  numbers  in  the  parentheses  indicate 
the  multiplicities  of  the  eigenvalues. 

The  interaction  space  in  this  degenerate  example  is  also  14-dimensional. 
Denote  Kj  =  FjKPj,  where  Pj  is  one  of  the  eight  projectors  defined  by 
equations  (11)-(16).  As  follows  from  the  discussions  of  Section  4,  spectral 
problem  (17)  can  be  decomposed  into  the  eight  problems 

{K,-XM)h  =  0. 

Factoring  the  matrices  Kj^  Kj  =  ajaj,  and  applying  equation  (8),  we 
arrive  at  the  eight  characteristic  matrices  Qj(X).  For  this  example,  Qj(X)  = 
I  -  (1/A)(7j,  where  qj  =  ajaj,  and  the  nonzero  eigenvalues  are  found  as 
solutions  to  the  eight  characteristic  problems:  {qj  —  X!)z  =  0. 

The  table  below  presents  the  dimensions  of  the  characteristic  matrices 
(the  third  row)  corresponding  to  each  of  the  projectors  defined  by  equa¬ 
tions  (11)-(16)  and  the  eigenvalues  obtained  by  solving  the  corresponding 
characteristic  problems. 


1512 


J 

1 

2 

3 

4 

5 

6 

7 

8 

Projector 

■Fb,i 

Po,2 

^3.1 

Psa 

Pho 

Pi,i 

^2,0 

■^2,1 

N, 

1 

0 

1 

1 

2 

2 

2 

2 

X 

2.000 

1.000 

3.000 

1.229 

4.271 

1.229 

4.271 

0.849 

2.651 

0.849 

2.651 

As  can  be  seen,  all  the  repeated  eigenvalues  of  the  original  problem  are 
split  now;  they  appear  cls  solutions  of  different  eigenvalue  problems  (one 
eigenvalue  corresponds  to  a  symmetric  vibration  with  respect  to  any  ver¬ 
tical  symmetry  plane  passing  through  the  6-fold  symmetry  axis,  and  the 
other  corresponds  to  the  antisymmetric  vibration).  Note  also  that  the  di¬ 
mension  of  the  characteristic  matrix  corresponding  to  the  projector  Po,2  is 
equal  to  zero.  This  implies  that  the  system  has  no  eigenvibration  such  that 
the  corresponding  vector  of  interaction  forces  belongs  to  the  space  Lo,2- 
In  conclusion,  note  that  though  the  decomposition  of  the  spectral  prob¬ 
lem  for  this  simple  example  results  in  no  advantages  (the  time  required  to 
factor  all  the  matrices  Kj  is  greater  than  the  time  required  to  solve  the 
original  problem  (17)),  the  situation  becomes  dramatically  different  in  the 
case  where  complex  subsystems  are  involved.  To  make  the  point  clear, 
let  us  imagine  that  Fig.  2  represents  the  interaction  section  for  a  system 
consisting  of  seven  complex  subsystems.  Let  circles  denote  the  connec¬ 
tion  points  of  the  subsystems.  Assume  that  the  Green’s  operators  of  the 
subsystems  are  available  (e.g.,  they  can  be  calculated  by  means  of  the 
modal  series).  If  one  applies  some  substructuring  method  for  investigation 
of  such  a  system,  one  arrives  at  the  investigation  of  a  certain  relatively 
small  matrix  (if  the  structural  analysis  method  [5,  6]  is  used,  the  order 
of  the  characteristic  matrix  will  be  equal  to  14),  which,  however,  depends 
nonlinearly  on  the  spectral  parameter  A.  The  finding  of  the  system  spec¬ 
trum,  in  this  case,  will  require  many  iterations  involving  calculation  of  the 
matrix  and  search  for  its  singularity  points  (zero  determinant).  The  use 
of  the  system  symmetry  in  this  case  will  allow  one  to  reduce  the  spectral 
problem  to  investigation  of  seven  characteristic  matrices  of  orders  one  or 
two  (see  the  above  table).  Since  the  skeleton  factorizations  of  the  matrices 
Kj  are  implemented  only  once  (before  the  iterations),  the  effect  of  using 
the  decomposition  is  evident. 

8  CONCLUDING  REMARKS 

The  method  of  investigation  of  complex  symmetric  structures  has  been 
presented.  The  technique  is  based  upon  results  of  the  group  representa¬ 
tion  theory.  To  obtain  the  decomposition  of  the  spectral  problem,  the 
interaction  space  of  the  system  is  decomposed  by  means  of  real  irreducible 
representations  of  the  group  describing  the  symmetry  of  the  system.  The 


1513 


projectors  onto  the  subspaces  of  the  interaction  space  are  given  for  the  two 
symmetry  groups. 

If  the  structure  under  investigation  possesses  the  symmetry  described 
by  the  group  (7„  or  Cnv,  in  order  to  decompose  the  spectral  problem,  one 
can  take  advantage  of  the  results  presented  in  the  paper.  In  this  case,  one 
needs  only  to  build  a  representation  of  the  group  in  the  relevant  interaction 
space.  The  way  of  construction  of  the  representation  of  the  group  C^y  for  a 
particular  configuration  of  the  interaction  points  is  presented  in  Section  6; 
it  can  be  easily  extended  to  other  configurations  that  fall  in  the  group  Cn 
or  Cnv 

As  to  other  kinds  of  symmetry,  a  general  scheme  of  application  of  the 
method  discussed  in  the  paper  can  be  briefly  summarized  as  follows: 

(i)  First,  the  symmetry  group  relevant  to  the  investigated  structure  is  de¬ 
termined. 

(ii)  Second,  the  connection  points  of  the  subsystems  are  separated  mental¬ 
ly  from  the  structure,  divided  into  independent  interaction  sections,  and  a 
representation  of  the  group  is  built  for  all  interaction  sections. 

(iii)  Then,  one  should  abstract  from  the  structure  and  to  find  the  decom¬ 
position  of  the  interaction  space.  This  is  achieved  through  finding  real 
irreducible  representations  of  the  group.  For  many  groups,  these  represen¬ 
tations  are  available  in  the  literature.  Note,  however,  that  they  are  often 
complex,  and  one  should  combine  pairs  of  complex  conjugate  irreducible 
representations  to  obtain  real  representations.  The  case  where  certain 
subspaces  transformed  by  representations  of  order  greater  than  one  can 
be  split  is  more  difficult  for  analysis.  To  determine  whether  or  not  this  is 
possible,  one  can  take  advantage  of  Theorem  2  or  try  to  find  other  criteria, 
which  may  occur  more  convenient  in  particular  cases.  This  stage  is  com¬ 
pleted  with  the  construction  of  the  projectors  onto  the  subspaces  of  the 
interaction  space.  The  projectors  are  built  by  means  of  formal  application 
of  formulas  (5)  or  (6)-(7). 

(iv)  Finally,  the  reduced  characteristic  matrices  are  constructed  by  means 
of  the  technique  described  in  Section  4. 

At  last,  note  that,  in  the  authors’  opinion,  the  approach  discussed  in  the 
paper  can  be  used  to  develop  a  method  for  analysis  of  another  interesting 
class  of  structures — the  structures  that  are  not  symmetric  but  consist  of 
identical  subsystems  (or  several  sets  of  identical  subsystems).  The  group 
relevant  to  such  a  class  of  systems  is  a  subgroup  of  the  permutation  group. 
We  suppose  that  the  close  examination  of  this  case  from  the  standpoint  of 
the  group  representation  theory  may  lead  to  the  development  of  an  efficient 
method  for  analysis  of  this  important  for  applications  class  of  structures. 


1514 


References 


1.  Bergman,  L.A.  and  McFarland,  D.M,  On  the  vibration  of  a  point 
supported  linear  distributed  structure.  J.  Vibr.,  Acoustics,  Stress  and 
Reliability  in  Design,  1988,  110,  485-492. 

2.  Garvey,  S.D.,  Friswell,  M.I.  and  Penny,  J.E.T.,  Efl&cient  evaluation 
of  composite  structure  modes.  Proc.  of  the  5th  Int.  Conf.  on  Recent 
Advances  in  Structural  Dynamics,  Southampton,  July  1994,  869-878. 

3.  Gould,  S.H.,  Variational  Methods  for  Eigenvalue  Problems,  Oxford 
University  Press,  London,  1966. 

4.  Kron,  G.,  Diakoptics,  MacDonald,  London,  1963. 

5.  Pesterev,  A.V.  and  Tavrizov,  G.A.,  Vibrations  of  beams  with  oscilla¬ 
tors  I:  structural  analysis  method  for  solving  the  spectral  problem.  J . 
Sound  Vibr.,  1994,  170,  521-536. 

6.  Pesterev,  A.V.  and  Tavrizov,  G.A.,  Structural  analysis  method  for  dy¬ 
namic  analysis  of  conservative  structures.  Int.  J.  Anal.  Exper.  Modal 
Analysis,  1994,  9,  302-316. 

7.  Pesterev,  A.V  and  Tavrizov,  G.A.,  Quick  inversion  of  some  meromor- 
phic  characteristic  matrices.  Proc.  of  the  5th  Int.  Conf.  on  Recent 
Advances  in  Structural  Dynamics,  Southampton,  July  1994,  890-899. 

8.  Pesterev,  A.V.  and  Bergman,  L.A.,  On  vibration  of  a  system  with  an 
eigenfrequency  identical  to  that  of  one  of  its  subsystems,  ASME  J. 
Vibr.  Acoustics,  1995,  117,  482-487. 

9.  Pesterev,  A.V.  and  Bergman,  L.A.,  On  vibration  of  a  system  with 
an  eigenfrequency  identical  to  that  of  one  of  its  subsystems,  Part  II, 
ASME  J.  Vibr.  Acoustics,  1996,  118,  414-416. 

10.  Yee,  E.K.L.  and  Tsuei,  Y.G.,  Direct  component  modal  synthesis  tech¬ 
nique  for  system  dynamic  analysis.  AlAA  J.,  1989,  27,  1083-1088. 

11.  Horn,  R.A.  and  Johnson,  C.R.,  Matrix  Analysis,  Cambridge  Univer¬ 
sity  Press,  Cambridge,  1985. 

12.  Lubarskii,  G.Y.,  Theory  of  Groups  and  its  Applications  to  Physics, 
Moscow,  1958,  (in  Russian). 


1515 


1516 


RANDOM  VIBRATION  I 


ANALYTICAL  APPROACH  FOR 
ELASTICALLY  SUPPORTED  CANTILEVER  BEAM 
SUBJECTED  TO  MODULATED  FILTERED  WHITE  NOISE 

Gongkang  Fu  and  Juan  Peng 
Department  of  Civil  and  Environmental  Engineering 
Wayne  State  University,  Detroit,  MI  48202,  USA 
TEL:(313)577-3842;  FAX:(3 13)577-3881 

SUMMARY 

For  elastically  supported  cantilever  beams,  this  paper  presents  an 
explicit  solution  of  response  statistics  to  seismic  input  at  the  base  -of  the 
beam.  The  excitation  is  modeled  by  modulated  filtered  white  noise.  The 
modulation  is  described  by  the  sum  of  exponentially  decaying  functions, 
accounting  for  nonstationarity  of  strong  ground  motion.  The  solution  was 
obtained  by  analysis  in  the  time  domain  using  the  state  space  approach  and 
impulse  response.  An  example  is  also  included  here  to  demonstrate  the 
application  to  bridge  piers,  in  developing  probabilistic  seismic  response 
spectra  for  design  and  retrofit  of  bridge  structures. 

1.  INTRODUCTION 

Although  various  methods  have  been  suggested  for  random  vibration 
problems,  only  a  few  explicit  solutions  have  been  reported  for  random 
vibration  under  nonstationary  excitations  Explicit  solutions  are 

often  desirable  for  their  advanced  abilities  of  verifying  approximation 
methods  and  providing  insights  of  the  problem.  This  paper  contributes  to  the 
knowledge  in  this  area  by  presenting  an  explicit  solution  for  the 
displacement,  velocity,  and  acceleration  of  a  structural  system  with 
continuous  parameters.  It  is  subjected  to  strong  ground  motion  modeled  by 
exponentially  modulated  filtered  white  noise.  The  double  filter  in  casecade 
suggested  by  Clough  and  Penzien  is  included  in  the  excitation.  This 
solution  was  obtained  by  analysis  in  the  time  domain  using  the  state  space 
approach  and  the  impulse  responses. 

2.  EQUATION  OF  MOTION 

Consider  a  vertical  cantilever  beam  with  constant  cross  section.  Its 
horizontal  displacement  V(x,t)  at  time  t  and  distance  x  from  its  base  is 
giverned  by  the  following  motion  equation,  with  an  assumption  of  viscous 
damping: 


1517 


m,  V”(x,t)  +c,  V’(x,t)  +  El  =  0 


(1) 


where  the  number  of  primes  to  V(x,t)  indicate  the  orders  of  partial 
derivatives  with  respect  to  time  t,  and  the  Roman  superscript  denotes  that 
with  respect  to  location  x  along  the  length  of  the  beam,  and  c^.  are 
constants  for  mass  and  damping  per  unit  length,  and  El  is  a  constant 
denoting  the  flexural  stiffness  of  the  beam. 

After  the  displacement  V(x,t)  is  decomposed  into  pseudo-static  and 
dynamic  displacements: 

V(x,t)=V3(x,t)  +  V,(x,t)  ;  V,  (x,t)=f(t)(l-x/L)  (2) 

Eq.(l)  becomes 

m,  V/’(x,t)  +  c,  V,’(x,t)  +  El  V;"(x,t)  =  -  m,  r(t)(l-x/L)  (3) 

It  is  assumed  here  that  the  damping  term  f  (t)(l-x/L)  is  negligible.  The 
associated  initial  and  boundary  conditions  are  identified  as  follows: 

v,(x,0)  =  0  v;(x,0)  ==  0  (4) 

k,  V,(0,t)  4-  El  V,>"{0,t)  =  0 
k.  v;(0,t)  -  El  V/(0,t)  =  0 
V/‘(L,t)  =  0 

V/(L,t)  =  0  (5) 

The  boundary  condition  at  the  base  (x=0)  in  Eq.(5)  describes  dynamic 
equilibrium  involving  deformations  at  the  base.  k(  and  k^  denote  the  elastic 
stiffness  for  translation  and  rocking,  respectively.  This  inclusion  is  intended 
to  model  effects  of  the  foundation.  For  example,  deep  pile  foundations  may 
be  modeled  by  setting  k^  and  k^  equal  to  infinity. 

The  ground  acceleration  is  modeled  by  a  nonstationary  random 
process,  described  by  the  product  of  a  deterministic  modulation  Tj(t)  and  a 
stationary  process  g(t): 

r(t)  =  ri(t)g(t)  (6) 

11  (t)  is  described  by  the  sum  of  a  series  of  exponential  functions  in  a  general 
form: 

Ti(t)=  Zi=,^,aiexp(bit)  (t>0;  ri(0)=0;  ri(oo)=0)  (7) 


1518 


where  a|  and  b;  are  real  constants,  and  bj  must  be  negative.  This  general 
form  of  modulation  is  used  here  for  including  several  popular  models  and 
ease  of  mathematical  derivation.  g(t)  is  assumed  to  be  a  white  noise  process 
with  zero  mean  and  filtered  by  the  Clough-Penzien  filter  modeling  soil 
effects  in  cascade.  Its  spectral  density  function  Sgg(co)  and  autocorrelation 
function  Rgg(x)  are  respectively  given  by 

0,/  +  (2i;g,0g:0)^  S 

Sgg(0)  - - (8) 

+  (2(;g,0g,0)"  (cOgj'-©^)^  +  (2Q2COg2to)'  271 
S  1 

Reg(x)=  -  Ik=i,2 - exp«g,0g, I x|) 

4  Cgic<^gk 

^gk^gk 

[(^ak  gdk  I  1  (Cak"Ct,(.)Sm0g(jV;  j  X  |  ]  (9) 

“gdk 

where  C,;,  and  C^k  (k=l,2)  are  respectively  as  follows; 

C,,  =  -  —  [(l+8<;„^-16Q/)(l-  — )  -  8^,,^ - (1-21;^^ - +  2?,,^ - )] 

D  COj,"*  co/  co/ 


20gi‘  “s'  “s'  “'6' 

Cg,  =  —  [(i+8;;gM6i;g,‘)(i-2^g,- —  +  2;;^,^— )-2q,^(i - )] 

D  aJ  “sz' 


kUg2  Wg!  v^gl  ^gl  ^gl 

Ca2  = - [(l  +  8Cgi^-16Cg,'^)(l — )  -  8^g,^— (l-2i;g2^-  —  +  2i;g,^ - )] 

D 


C0g2^  C0g2' 

©g2'  C0g2' 

®gl' 

®gi" 

—  (- 

— -8Cg,^+l6gV)(l-2^;s.- 

—  +  2g^ — ) 

(»g2' 

^g2^ 

C0e2'  C0g2- 

— )] 

(10a) 

®g2' 

and 


1519 


(10b) 


®Sl 

D  =  -40),,^(0,,^(l-2^„^ - +  - + 

0)/  CB,j"  (0,j"  O5/ 

+  (0/(l - )^ 

In  Eqs.(8)  to  (10),  (^g,,  ^gj,  cOg,,  and  C0g2  are  characteristic  damping  ratios  and 
frequencies  of  the  respective  subfilters  indicated  by  subscript  k=l,2.  cOg^k 
=  cDgk(l-(!^gk^)'^  (k=  1,2)  are  their  characteristic  damped  frequencies.  8/271  is 
a  constant  indicating  the  intensity  of  white  noise. 

Note  that  RggCr)  is  the  Fourier  inverse  transform  of  Sgg(cD)  : 

^£8^^)  exp(/coT)  dco  (11) 

where  /=(-l)‘'^  is  the  imaginary  unit  and  Rgg(T)  was  obtained  by 
decomposing  Sgg(co)  into  the  sum  of  two  terms: 

S  Cak  COgk^  +  CbkCD^ 

Sgg(o))=  —  Ik=t.2 -  (12) 

27:  (o)gi^-coy  +  (2(;g,0)g,CD)" 

and  performing  the  inverse  Fourier  transform  individually  for  each  term  by 
the  method  of  residues  Note  that  the  Kanai-Tajimi  filter  can  be 
represented  in  Eqs.(8)  to  (10)  by  setting  C0g2=0,  leading  to  Ca,=C0gj^, 
and  Ca2=Ci,2=0  On  the  other  hand,  C0gi->oo  eliminates  the 
effect  of  the  Kanai-Tajimi  filter.  Note  that  this  seismic  excitation  model  of 
modulated  cascade-filtered  white  noise  covers  nonstationarity  of  intensity, 
but  not  that  of  frequency  contents.  Either  or  both  of  them  may  be  observed 
in  strong  motion  records. 

3.  SOLUTION  BY  STATE  SPACE  FORMULATION 
3.1  Modal  Decoupling 

Eq.(3)  is  solved  here  by  separating  the  two  variables  x  and  t,  and 
using  modal  superposition: 

V,(x,t)  =  Zi.,.2.,..<i>i(x)Vi(t)  (13) 

where  (j);  (x)  is  the  ith  mode  shape  and  Vi(t)  is  the  normal  coordinate  for  that 


1520 


mode.  Using  modal  orthogonality,  Eq.(3)  becomes  a  series  of  decoupled 
equations  of  modal  motion: 

v,”(t)  +  2  Q  CO,  V,>(t)  +  CO,'  V,(t)  =  S,  f’(t)  (14) 


where 

cOi'  -  a^'  El/m, 

Ci  =  c,/(2  coim,) 

Si  ==  (1-x/L)  dx  /  Io‘  (j);  ^(x)  dx  (i  =  l,2...)  (15) 

(Oj  and  Q  are  respectively  modal  natural  frequency  and  damping  ratio.  S;  is  a 
factor  indicating  the  extent  of  participation  of  the  mode.  The  mode  shapes 
are  given  as 

(l)i  (x)  =  Cl  i  sin(aiX)  +  C2,i  cos(aiX)  +  C3  ^  sinh(a|X)  +  C4  i  cosh(aiX)  (16) 

a;  related  to  the  natural  frequency  is  the  solution  to  the  eigen  value  problem, 
and  the  coefficients  of  c,  ■  to  04,;  are  obtained  by  meeting  the  boundary 
condition  in  Eq.(5)  for  every  mode. 

3.2  Modal  Solution  in  State  Space 

Let  the  state  variables  for  mode  i  be  denoted  by 

Vi(t)  =  (Vi(t).  V,’(t),  Vi’Tt))'^  (17) 

where  superscript  T  denotes  transpose  of  matrix.  The  inclusion  of  Vi”(t)  is 
to  provide  a  complete  picture  of  the  system  behavior,  and  to  be  able  to 
describe  the  maxima  of  displacement  as  discussed  later. 

The  solution  to  Eq.(14)  can  be  shown  in  the  following  convolution 

form: 

Vi(t)  =  -  Si  C|  1  o‘  Zj  (t-x)  f’(x)  dx  (18) 

where  C;  is  a  matrix  of  system  parameters  for  mode  i,  and  Zi(t)  is  a  time- 
dependent  vector: 

(  0 

Ci  =  I  1 


1/cOdi  0) 

-Q  cDi/cOdi  0 1  (19a) 

(^i  l^* 


1521 


(19b) 


Zi(t)  =  (exp(-  ©it)  cos  ©dit.  exp(-  ©jt)  sin  ©^jt,  SCt))"^ 


where  ©^j  =  ©i(l-^  and  5(t)  is  the  Dirac  delta  function. 

Since  the  ensemble  expectation  of  Vi(t)  is  a  zero  vector  due  to  zero 
mean  of  the  input  process  g(t),  the  covariance  matrix  of  Vi(t)  is 

=  E[Vi(t)  v,^(t)]  =  S;^  Ci  B|(t)  C,^ 


Bi(t)  =  Jo'Jo'T|(T,)r|(-Cj)Rj5(|T,-tj|)Zi(t-T,)Zi'^(t-t2)<i'Ci(lt2  (20) 

where  E  stands  for  ensemble  expectation.  This  double  integration  has  been 
performed  as  shown  in  by  analysis  in  the  time  domain. 

It  will  be  seen  below  that  the  covariance  matrix  for  two  different 
modes  (say  m  and  n)  is  needed  for  further  analysis.  This  matrix  can  be 
similarly  expressed  as: 

R.,,„(t)  =  EK(t)  v^-^ft)]  =  S„  B,„(t)  C  J 

Bmn(t)  =  lo%'^(^l)Tl('C2)RgE(Kr'r2l)Zm(t-'^l)ZnVT:2)dTidx^  (21) 

This  integration  is  also  carried  out  in  the  time  domain.  The  results  of  B^n(t) 
are  explicitly  given  in  Appendix.  Note  that  when  m=n=i,  B„n(t)=  Bi(t). 
Eqs.(20)  and  (21)  show  that  the  statistics  of  the  nonstationary  responses  are 
expressed  as  product  of  a  stationary  part  (constant  Cj  matrix)  and  a 
nonstationary  part  (matrix  Bi(t)  or  B„n(t)). 

3.3  Mean  Square  Total  Response  in  State  Space 

With  the  elementary  terms  given,  the  covariance  matrix  Rvv(t)  of  the 
vector  of  dynamics  state-variables  v(t)  can  be  now  formulated  as  follows: 

v(x,t)  =  (V,(x,t),  V,(x,t),  V,”(x,t))’' 

=  (i:i=i,2....cl>.(x)  V,(t),  Zi=,2....<l)i(x)Vi’(t),  i:i=,2..,.<^.(x)Vi”(t))  (22) 

R,,(t)  =  E[v(x,t)  v'^(x,t)] 

=  In.l.2,...^.n(x)S,  (l)„(x)S„  C,  B„,(t)  Cj 

f  Rvv.ll(X,t)  Rvv,,2(x,t)  Rvv.l3(X,t)  "l 

=  i  R,v.2l(X,t)  Rvv,22(X,t)  Rvv.23(X,t)  1  (23) 

^Rvv.3!(X,t)  Rvv.32(X,t)  R,,.33(X,t)j 

Note  that  the  second  group  of  subscripts  indicates  the  physical  quantities  by 


1522 


1=  displacement,  2= velocity,  and  3  =  acceleration.  This  system  of 
identification  will  be  used  hereafter  for  variances.  Of  the  summed  terms  in 
Eq.(23),  the  dominant  terms  are  often  those  referring  to  the  same  mode,  i.e. 
((^,(x)S„)^  CA(t)C7(n=l,2,...). 

4.  NUMERICAL  EXAMPLES 
4.1  Unit  Step  Modulation 

The  modulation  function  given  in  Eq.(7)  is  in  a  general  form  to 
describe  nonstationarity  of  earthquake  excitations.  It  can  be  reduced  to 
several  commonly  used  models.  For  example,  the  unit  step  function  model 
of  modulation  is  a  special  case  represented  by  N  =  l,  ai  =  l,  and  bj=0  in 
Eq.(7).  This  model  is  perhaps  the  earliest  modulation  envelope  for  modeling 
nonstationarity  of  earthquakes  When  effects  of  the  filter  is  eliminated  by 
setting  (X)gi->oo  and  C0g2=0,  the  solution  in  Eq.(20)  for  a  mode  reduces  to  the 
response  of  an  oscillator  to  white  noise  modulated  by  the  step  function,  with 
natural  frequency  (Hq,  damping  and,  of  course,  the  modal  participation 
factor  So  =  l  in  Eq.(14): 

S  exp(-2(;oCOot) 

Rvvti(0= - {1+ - [  -  l+^;o^cos2c0dot-(;o(l-^o^)‘^sin2co^ot]}  (24) 

4CoCio'  (1-Co^) 

S 

Rvv  p(t)  = - exp(-2(;oC3ot)(  1  -  cos2c0dot )  =  Rvv,2i(f)  (25) 

4(Oo^(l-Co^) 


S  exp(-2i;oCOot) 

Rvv  22(t)  = - {  1  + - [  -1  +^o'cos2co,ot+^(l-Co')""sin2co,ot]}  (26) 

4(;o0)o 

s  exp(-2(:;otoot) 

Rvv  ,3(0  - - {  -1  + - [l-2(:;o'+Co'cos2o3,ot+(;o(l<oO'''sin2o)dot]} 

4^ot0o  d-CoO 

-  Rvv.3,(0  (27) 

S 

Rvv  23(0  = - exp(-2(;oCOot)[  1  +  (l-2^o0cos2o3,ot -2^o(l- ^o0'^'sin2co,ot] 

4(1- 


1523 


=  Rvv.32(t) 


(28) 


Rvv.33(0  — ^  °°  (29) 

The  first  three  terms  have  been  given  elsewhere  The  last  term  is 

contributed  by  the  white  noise  acceleration  input,  representing  an  impulse  of 
infinite  peak  at  time  lag  t:=0. 

4.2  Shinozuka-Sato  Modulation 

Another  group  of  modulation  models  is  described  by  the  Shinozuka- 
Sato  model  which  is  a  special  case  of  N==2  in  Eq.(7).  Examples  are 
a,  =2.32,  a2=-2.32,  b,=-0.09,  and  b2=-1.49  for  El  Centro  earthquake,  and 
a,  =  12.8,  a2=-12.8,  bi=-0.14  ,  and  b2=-0.19  for  Taft  earthquake  The 
latter  is  used  in  this  example,  with  cOgj  =  6.47:  rad/s,  =  0.65,  C0g2  = 
0.3271  rad/s,  and  =  0.5 

For  a  typical  reinforced  concrete  pier  of  highway  bridges,  L=12m, 
EI/n\  =  250  kN/kg-m\  and  ^,=0.04  (i=l,2,...)  are  used  here.  The  constant 
damping  is  used  because  it  is  felt  that  the  one  inversely  proportional  to  the 
modal  frequency  as  defined  in  Eq.(15)  is  not  realistic  for  concrete  structures. 
Three  sets  of  stiffness  ratio  rt=ktLVEI  and  rf=  k^L/EI  are  used  here:  q  ->  oo 
and  r,^  oo,  q  =1  and  r=l,  and  r=0.1.  The  variances  of  displacement, 
velocity,  and  acceleration  at  the  top  (free)  end  of  the  beam  (x=L)  are  shown  in 
Figs.l  to  3  for  the  three  cases,  respectively.  Unit  intensity  is  used  here,  coj  is 
the  first  natural  frequency  of  the  beam.  It  is  seen  that  decrease  of  the 
foundation  stiffness  causes  the  response  to  increase,  showing  the  importance 
to  take  into  account  the  foundation  stiffness.  They  also  show  that  the  peak 
responses  are  slightly  delayed  by  lower  foundation  stiffness.  On  the  other 
hand,  additional  analysis  results  show  that  the  internal  forces  (shear  and 
moment)  are  reduced  by  lower  foundation  stiffness. 

4.3  Application  to  Probabilistic  Response  Spectrum 

The  mean  square  response  statistics  shown  here  may  be  used  to 
develop  probabilistic  seismic  response  spectra  for  interested  failure  modes. 
An  example  is  given  here.  Consider  a  bridge  span  with  one  end  supported 
by  an  abutment  with  fixed  bearings  and  the  other  by  a  pier  with  expansion 
bearings.  The  fixed  bearings  and  the  bridge  deck  (superstructure)  are 
modeled  as  an  SDOF  system  with  mass  M,  damping  C,  and  stiffness  K 
subjected  to  the  strong  motion  acceleration  f’(t)  through  the  abutment: 

Mw”(t)  +  Cw’(t)  +  Kw(t)  =  -Mf”(t)  (30) 


1524 


where  w(t)  is  the  horizontal  displacment  of  the  deck  relative  to  the  abument. 
The  relative  displacment  u(t)  at  the  end  supported  by  the  pier  is  focused 
here: 

u(t)=  w(t)- V(L,t)  (31) 

where  u(t)=(u(t),u’(t),u”(t))'^.  It  is  critical  to  understand  the  dynamics  of 
the  system  when  designing  the  bridge  to  cover  the  failure  mode  of  span 
collapse  due  to  excessive  relative  displacement  u(t).  The  covariance  of  u(t) 
can  be  readily  expressed  as 


^uu(0  —  Rww(0  "  ^wv(0  ■  Rvw(0  Rw(l) 

Rvv(t)  has  been  given  in  Eq.(23),  R^,,(t)  can  be  expressed  using  Eq.(20), 
R,^v(0  and  Rv„.(t)  can  be  readily  formulated  using  Eq.(21)  and  taking 
summation  over  the  significant  modes  involved  in  vector  V(t).  Note  that  the 
length  of  cantilever  beam  L  in  Eq.(31)  is  omitted  in  Eq.(32). 

Now  let  us  define  the  failure  of  excessive  relative  displacement.  Let 
E[M(-co,t)]  be  the  mean  rate  of  displacement  maxima  at  time  t  and 
E[M(U,t)]  be  the  mean  rate  of  displacement  maxima  above  a  given  level  U, 
they  are  respectively 

E[M(-oo,t)]=  -  L"duL°  u”  p,,.  „>(u,0,u”,t)du”  (33) 

E[M(U,t)]  =  -  Iu"duL°  u”  „..(u,0,u”,t)du”  (34) 

where  p^  ,,- ^•■(.)  is  the  joint  probability  density  function  for  u,  u’,  and  u”.  If 
g(t)  in  Eq.(6)  is  assumed  to  be  a  Gaussian  excitation,  these  state  variables 
are  also  Gaussian  variables.  E[M(-co,t)]  and  E[M(U,t)]  are  then  derived  as: 

?^'(t)  Ruu.33'^^(t)  1 

E[M(-oo,t)]  = -  (35) 

27:  K..22%)  1“X'(0 

U 

E[M(U,t)  =  E[M(-«5,t)]  {0( - ) 

Mt)  Ruu.ii'^'(t) 

U' 

-  x(t)  exp[ - (l-x^(t))]  0(- 

2X^(t)  R,,H(t) 


X(t)U 

- ) }  (36) 

X(t)  R,,,n^^^(t) 


where 


1525 


|R(t)l 

X^(t)  = - 

^u,ll(0  ^u.22(0  ^u,33(0 

X(t)  =  y,3(t)  -  Yl2(t)  Y23(t) 

Ymn(t)  =  Ruu.mn(t)/[I^u.rnm(t)R^u.nn(0]’^  (m,n=  1 ,2,3 ,  HI  II)  (37) 

|R(t)|  is  the  determinant  of  R„„(t),  and  0(.)  is  the  cumulative  probability 
function  for  the  standard  normal  variable. 

Let  probability  of  failure  due  to  excessive  displacement  be 

Pf  =  Probability  [  maximum  u  >  U  ] 

=  E[M(U,t)]  dt  /  E[M(-oo,t)]  dt  (38) 

where  Tj  is  the  interested  time  length,  being  the  time  interval  of  significant 
seismic  input.  Pj-  indicates  the  likelihood  that  peak  displacements  exceed  a 
given  level  U.  For  a  given  Pf  and  seismic  input  intensity  S,  variation  of  the 
threshold  level  U  as  a  function  of  structural  system  frequency  and  damping 
is  defined  here  as  probabilistic  displacement  response  spectrum,  because  it  is 
associated  with  a  probability  to  be  exceeded.  This  spectrum  can  be  used  for 
design  to  control  displacement.  For  example,  the  minimum  seat  width 
requirement  for  highway  bridges  subjected  to  seismic  hazard  may  be  derived 
using  this  approach  for  risk-based  design.  For  the  example  of  bridge  pier 
discussed  above,  it  is  found  that  when  the  pier’s  first  natural  frequency  is 
close  to  that  of  the  superstructure-bearing  system,  U  may  be  significantly 
lower.  This  is  because  the  two  systems  behave  very  much  similarly  due  to 
their  similar  dynamic  properties  so  that  their  relative  displacement  u(t) 
approaches  zero.  Note  that  this  concept  of  probabilistic  response  spectra  can 
be  applied  to  acceleration  for  force  control  design. 

5.  CONCLUSIONS 

An  explicit  solution  is  presented  in  this  paper  for  random  vibration  of 
elastically  supported  cantilever  beams  subjected  to  the  strong  ground  motion 
modeled  by  modulated  cascade-filtered  white  noise.  It  may  be  used  to 
develop  probabilistic  response  spectra  for  risk-based  seismic  design  and 
retrofit  specifications. 


REFERENCES 

[1]  G.Ahmadi  and  M.A.Satter  "Mean-Square  Response  of  Beams  to 
Nonstationary  Random  Excitation",  AIAA  Journal,  Vol.l3,  No.8,  pp.l097- 


1526 


1100 

[2]  C.G.Bucher  "Approximate  Nonstationary  Random  Vibration  Analysis 
for  MDOF  Systems",  J.App.Mech.  Vol.55,  pp.197-200,  1988 

[3]  T.K.Caughey  and  H.F.Stumpf  "Transit  Response  of  A  Dynamic  System 
under  Random  Excitation,  J.  Appl.  Mech.,  ASME,  Vol.28,  pp. 563-566, 
1961 

[4]  R.W. Clough  and  J.Penzien.  Dynamics  of  Structures,  McGraw-Hill 
Book  Company,  1975 

[5]  R.B.Corotis  and  T. A. Marshall.  "Oscillator  Response  to  Modulated 
Random  Excitation, "  ASCE  J.Eng.Mech.  Vol.103,  EM4  pp. 501-5 13,  1977 

[6]  G.Fu  "Seismic  Response  Statistics  of  SDOF  System  to  Exponentially 
Modulated  Coloured  Input:  An  Explicit  Solution”  Earthquake  Engineering 
and  Strucmral  Dynamics,  Voi.24,  1995,  pp.  1355-1370 

[7]  D.A.Gasparini  "Response  of  MDOF  Systems  to  Nonstationary  Random 
Excitation",  ASCE  J.  of  Engg.  Mech.,  Vol.105,  No. EMI,  Feb.  1979, 
pp. 13-27 

[8]  D.A.Gasparini  and  A.DebChaudhury  "Dynamic  Response  to 
Nonstationary  Nonwhite  Excitation"  ASCE  J.Eng.Mech.  Vol.106,  EM6, 
pp. 1233-1248,  1980 

[9]  Z.K.Hou  "Nonstationary  Response  of  Structures  and  Its  Application  to 
Earthquake  Engineering",  California  Institute  of  Technology,  EERL  90-01, 
1990 

[10]  W.D.Iwan  and  Z.K.Hou  "Explicit  Solution  for  the  Response  of  Simple 
Systems  Subjected  to  Nonstationary  Random  Excitation",  Structural  Safety, 
Vol.6,  1989,  pp.77-86 

[11]  K.  Kanai  "Semi-empirical  Formula  for  the  Seismic  Characteristics  of 
the  Ground",  Univ.  of  Tokyo  Bull.  Earthquake  Res.  Inst.,  vol.  35,  pp.  309- 
325,  1957 

[12]  Y.K.Lin.  Probabilistic  Theory  of  Structural  Dynamics,  R.E.Krieger 
Publishing  Company,  1967 

[13]  M.Shinozuka  and  Y.Sato  "Simulation  of  Nonstationary  Random 
Process",  ASCE  J.Eng.Mech. Div.,  Vol.93,  pp.11-40,  1967 

[14]  M.Shinozuka  and  W-F.Wu  "On  the  First  Passage  Problem  and  Its 
Application  to  Earthquake  Engineering",  Proc.  9th  WCEE,  Aug. 2-9,  1988, 
Tokyo-Kyoto,  Japan,  P.VIII-767 

[15]  H.Tajimi  "A  Statistical  Method  of  Determining  the  Maximum  Response 
of  a  Building  Structure  during  an  Earthquake",  Proc.  2nd  World  Conf. 
Earthquake  Eng.  Tokyo  and  Kyoto,  vol.II,  pp.781-797,  July  1960. 

[16]  C.-H.  Yeh  and  Y.K.Wen  "Modeling  of  Nonstationary  Ground  Motion 
and  Analysis  of  Inelastic  Structural  Response"  Structural  Safety,  Vol. 8, 
1990,  pp.281-298 

Appendix 


1527 


(Al) 


f  ®mn.ll  ^mnJ2  ®mn,13 
®mn(0  1  ®mn,2I  ®mn,22  ®mn,23 

^  ®mn,3I  ®mn,32  ®mn.33 


B.n.n(t)  =  S  eXp(-0)„^„-C0„(;n)t  -  2i.,.N  Ij=,.N  aiHj  * 

^^^glc®gk 

{(Cak + c  J  [g  1  (akn„a,„,  Pj,„,  ^i,„)-g2(ai^,  p  kniPjlansP-ikn) 

§3(^kiti>  ^kn»  Pjkm » Pikn)“§4(^  km  ’  Pkn » Pjkm » Pikn) 

S 1  (  P  km  ’  Pkn  >  Pjknn  Pikn)  ”§2  (Pkm  >  ^kn » Pjkm  5  Pikn) 

§3  (  P  km » Pkn » Pjkm  >  Pikn)  ~S4(  Pkm  ’  ^kn*  Pjkm  ’  Pikn) 

S5(^km5  ^ktn  Pjkm»  Pikn)“S6(^km’  Pkm  Pjkm’  Pikn) 

+  g7(akm>a 

kn»Pjkm>  Pikn)~S8(^km»  Pkn’Pjkm’  Pikn) 

*^g5(Pkm’Pkn»Pjkm>Pikn)“g6(Pkm>^kn»Pjkm»Pikn) 

+  g7(Pkm>Pkn»Pjkm,Pikn)-g8(P  km » ^kn  ’  Pjkm » Pikn) 
~^Cgk®gk^®gdk(^ak”^bk)  [g9(^km >  ^kn » Pj km >  Pikn) ~glo(^km>Pkn>Pjkm’ Pikn) 

+gii(akm>a  kn  5  Pjkm  >  Pikn)‘g  1 2  (*^km  >  P  kn » Pjkm  ’  Pikn) 
~g9(Pkra’Pkn»Pjkm’Pikn)”^glo(Pkm’^kn’  Pjkm  ’  Pikn) 
"gil(Pkm>Pkn’Pjkm>Pikn)"^gl2(Pkm’^kn»  Pjkm » Pikn) 
gl  sC^km  ’  ^kn»  Pjkm  ’  P ikn)”g  14(^km  >  Pkn  >  Pjkm » Pikn) 
“^gl5(*^km>*^kn>Pjkm»Pikn)“gl6(^km»Pkn’PjkmJpikn) 
“glS^Pkm’Pkn’Pjkm’Pikn)  "^gl4(Pkm ’*^kn’ Pjkm’ Pikn) 
“gl5(Pkm’Pkn’Pjkra’Pikn)  gl6(Pkm’^kn’ Pjkm’ Pikn) 


Bmn.l2(0  —  Bnm.2l(0 

1 

B„,n.i3(0  =  S  [Zi=,,,s,  a,  exp(bit)]  I, =1.2 -  aj  exp(bjt)  * 

S^gk^gk 

{(^ak“^^bk)[gl8(^  km’  Pjkm)  '^gl8(P  km’  Pjkm)] 
■*"^gk®gk^®gdk(Cak"^bk)["gl7(^km’Pjkm)"*"gl7(Pkm’Pjkm)]} 


1 

B„n,2i(t)  =  S  exp(-®^i;„-co„Qt  I, ^,  2 - 2^i=i.N  Ij=i,N  Siaj 

16Cgk<«gk 

{  (^ak  ^bk)  ["g9(^km  ’  ^kn  ’  Pjkm  ’  Pikn)  g  1  o(^km  ’  P  kn  ’  Pjkm  ’  Pikn) 

~g  1 1  (^km  ’^kn  ’  Pjkm  ’  Pikn)  g  1 2 (®'km  ’  P kn  ’  Pjkm  ’  Pikn) 
”g9(Pkm  ’  Pkn»  Pjkm’  Pikn)  g  lo(Pkm’^kn’  Pjkm  ’  Pikn) 

~gl  l(Pkm’Pkn ’Pjkm’ Pikn)  '^g|2(P  km’  ^kn’ Pjkm ’Pikn) 

"b  g  1 3  (Ct  itm  ’  ^kn  ’  Pjkm  ’  P  ikn)“g  1 4(^km  ’  Pkn  ’  Pjkm  ’  Pi  kn) 

“b  g  t5(^km  ’^kn’  Pjkm’  Pikn)“gl6(^km’  Pkn’  Pjkm’  Pikn) 

+  gl3(Pkm’P  kn’ Pjkm’ Pikn)  gl4(Pkm’^kn’  Pjkm’ Pikn) 


(A2) 

(A3) 

(A4) 


1528 


"^SlsCPkm’  Pkn»!^jkm>Pikn)”§16(P  km  ’  *^kn  ’  P'jkm » P  ikn) 

^gk®  gk^  ®  gdk  (^ak”^bk)  [Sl(^km’^kn’Pjkm»  M'ikn)~S2(*^km  >Pkn’Pjkm»Pikn) 

§3(*^km5^kni  Pjkm’  Pikr)“S4(^km»  Pkn’  Pjkm’P'ikn) 

“§  1  (  Pkm  ’  Pkn  >  Pj  km  ’  M-ikn)  S2(P  km » *^kn  >  Pjkm » M-ikn) 

"SsCP  km  >  Pkn  ’  Pjkm » M-ikn)  §4(P  km  ’  ^kn  *  Pjkm  >  l^ikn) 

"S5(*^km5 ^kfijM-Jlcm’ Pikn)  §6(^km>  Pktu  M-jkm’ Pikn) 
-g7(ak^,a;,„,^jkm.Pikn)+g8(Ct  km  >  Pkn  >  M’jkm  ’  Pikn) 
g5(Pkm’  Pku’  M-Jkm*  Pikn)~S6(Pkm»^kn>  Pjkm>  Pikn) 

S7(Pkm » Pktu  Pjkm » Pikn)"g8(Pkm  >  *^kn?  Pjkm»  Pikn)  (-^5) 

1 

®mn,22(0  ~  ^  eXp(-COn,!l^n;,-CDj,(!^j,)t  2  ^i  =  l,N  ^j  =  l,N  ^i^j 

16Cgk®gk 

{  (^ak  ^bk)  [g  1  (^km  »^kn’Pjkm’Pikn)'^g2(^km>Pkn’Pjkm»Pikn) 
g3(*^km  ’  ^kn  ’  Pjkm  >  Pikn)  g4(*^km  >  Pkn  J  Pjkm  ’  Pikn) 

”^gl(Pkm»Pkn’Pjkm»Plkn)”^g2(Pkm’^kn»Pjkm5pikn) 
g3(Pkm»  Pkn»  Pjkm’Pikn)  g4(Pkm»*^  knJpjkm>Pikn) 

+  g5(Ctkm>CX,„,^jkm.Pikn)  +  g6(a  km’Pkn»Pjkm>Pikn) 
g7(^km  >  ’^kn » Pjkm  ?  Pikn)  g8(^km  >  Pkn » Pjkm  ’  Pikn) 

“^gsCPkm’  Pkn»Pjkm>Pikn)  "^ge^P  km » ^kn  ’  Pjkm  5  P  ikn) 

■^g7(Pkm’Pkn'! Pjkm’ Pikn)  “^g8(Pkm»^kn>Pjkm5pikn)] 

^gk^gk^^gdk(^ak~^bk) [g9(^km’*^kn’  Pjkm ’  Pikn)  glo(^km’  Pkn’  Pjkm?  Pikn) 
g  n  (^km  ’  ^kn  ’  Pjkm  ’  Pikn)  g  I2(^km  ’  Pkn  ’  Pjkm » Pikn) 

■g9(Pkm>Pkn’Pjkm>Pikn)"glo(Pkm»^kn’Pjkm’Pikn) 
"gH(Pkm’Pkn’Pjkm’Pikn)“gl2(Pkm’^kn’ Pjkm’ Pikn) 

+  gl3(akm’akn’Pjkm’Pikn)  +  gl4(akm’  Pkn’ Pjkm’ Pikn) 

gl5(^km’*^kn’ Pjkm’ Pikn)  "^gieC^km’ Pkn’ Pjkm’ Pikn) 


"gl3(Pkm’Pkn»Pjkm’Pikn)"gl4(Pkm’^kn’ Pjkm’ Pikn) 

■g  IsCPkm  ’  Pkn’  Pjkm  ’  Pikn)“g  leCPkm’^kn’Pjkm’  Pikn)] 

(A6) 

1 

^mn,23(0  ~  S  [I]j=i,N  ^i  ®^P(bit)]  I]ic  =  l,2  Sj  =  l.N  ^j  ®XP(t>jt) 

^^gk®gk 

{  (^ak  ^bk)  [gl7(^km ’  Pjkm)  gl7(Pkm >  P jkm)] 

*^gk®  gk^^gdk(^ak“^bk)  [gl  S^^^km  ’  Pjk)"gl  sCPkm  ’  Pjkm)!  } 

(A7) 

Smn,3l(0  =  Bnn,  13(1) 

(A8) 

®mn.32(0  Bnm.lsCO 

(A9) 

B^n.33(t)  =  E=i.n  a;  exp(bit)]'  Rgg(O) 

(AlO) 

1529 


Vnriance 


where  P,,.  and  p,„  are  totions  of  system  parameters  givea  below; 


CL^  =  -  CiDjdk  s  •  Pier  -  ^dn  “  ’ 

Pjerr  =  bj  “b  ’  P-ilcn  ” 

(ij  =  l,2 . N;  k=l,2;  m,n=l,2,...) 

Note  chat  fanctions  g,  through  gt8  are 


-h  -  Cj,C0j, 


(All) 


not  siven  here  due  to  limited  space. 


Fig.l  Variance  of  Top  Displacement,  velocity,  and  Acceleration 
(S  =  1 ,  r,  -»  and  ®) 


1532 


LINEAR  MULTI-STAGE  SYNTHESIS 
OF  RANDOM  VIBRATION  SIGNALS 
FROM  PARTIAL  COVARIANCE  INFORMATION 


S.D.  Fassois*  and  K.  Denoyer** 

*Deparl-,menl'.  of  Mechanical  &  Aeronautics  Engineering 
University  of  Patras 
GR-265  00  Patras,  Greece 
E-mail:  fassois@mech.upatras.gr 

**Department  of  Mechanical  Engineering  &  Applied  Mechanics 
The  University  of  Michigan 
Ann  Arbor,  MI  48109-2125,  U.S.A. 


ABSTRACT 

An  effective  scheme  for  random  vibration  signal  synthesis  from  partial  co- 
variance  information  is  introduced.  The  proposed  scheme  is  based  upon 
a  Fast  Rational  Model  estimation  approach,  combined  with  a  discrete 
ARMA(2n,2n-l)  vibration  signal  representation  and  a  dispersion  analysis 
methodology.  Unlike  previous  approaches,  the  proposed  scheme  provides 
accurate  synthesis  without  resorting  on  nonlinear  operations,  and  provides 
the  tools  for  effective  representation  order  determination.  Rirthermore,  it 
mathematically  guarantees  the  estimated  representation  stability  and  in- 
vertibility  properties,  implying  that  all  types  of  vibration  signals,  including 
those  characterized  by  sharp  spectral  peaks  or  valleys,  can  be  handled  with¬ 
out  difficulty.  The  effectiveness  of  the  scheme  is  demonstrated  via  synthesis 
test  cases,  through  which  the  benefits  of  representation  overfitting  are  also 
presented. 


G{s) 

h 

S[u) 

7(t) 

7[/cri 

4>[B) 

6{B) 

4>i 

Oi 

t-H 


List  of  Symbols 
receptance  transfer  matrix 
j-th  term  of  the  Green’s  function 
j-th  term  of  the  inverse  function 
power  spectral  density  function 
continuous  time  auto  covariance  function 
discrete  time  auto  covariance  function 
autoregressive  polynomial 
moving  average  polynomial 
i-th  autoregressive  parameter  (i  =  1, 2,  •  •  ■ ,  2n) 
i-th  moving  average  parameter  {i  =  1, 2,  •  •  * ,  2n  -  1) 
■i-th  continuous  pole  (i  =  1, 2,  •  •  • ,  2n) 


1533 


5i  ;  z-th  vibration  mode  dispersion  percentage  (i  =  1, 2,  •  •  •  ,n) 
cr^  :  innovations  variance 
B  :  backshift  operator  (J5rr[i]  =  —  1]) 

R  :  number  of  specified  autocovariance  samples 

T  :  sampling  period 

p  :  truncated  inverse  function  order 

no  :  number  of  structural  degrees  of  freedom 

n  ;  number  of  degrees  of  freedom  in  the  signal  representation 

Superscript: 

^  :  indicates  iteration  number. 

Conventions: 

[•]  :  function  of  a  discrete  variable. 

(•)  :  function  of  a  continuous  variable. 

Vector/matrix  quantities  are  indicated  in  bold-face  characters. 


1.  INTRODUCTION 

In  this  paper  the  problem  of  digitally  synthesizing  stationary  random 
vibration  signals  from  partial  (incomplete)  covariance  information  is  con¬ 
sidered.  The  problem  is  important  for  generating  vibration  signals  to  be 
used  in  the  computer  or  laboratory  based  simulation  and  testing  of  various 
types  of  structural  systems,  including  machine,  vehicle,  aerospace,  and  civil 
engineering  structures. 

The  problem  of  equi-spaced  in  time  vibration  signal  synthesis  from 
complete,  analytically  available,  covariance  information  may  be  tackled  via 
well  known  spectral  factorization  techniques  [1],  which  yield  an  appropriate 
stochastic  realization  that  may  be  subsequently  used  for  synthesis  purposes. 
The  practically  much  more  important  problem  of  synthesis  based  upon  a 
finite  (truncated)  number  of  available  covariance  samples  is  equivalent  to 
that  of  finding  an  admissible  extension  of  the  partially  specified  covariance 
sequence,  and  obtaining  an  analytical  description  that  can  be  used  for 
vibration  synthesis.  A  complete  analytic  description,  which  automatically 
implies  an  admissible  extension  of  a  given  covariance  sequence,  as  long  as 
the  latter  corresponds  to  a  rational  spectral  density,  is  provided  through 
the  AutoRegressive  Moving  Average  (ARMA)  representation  [2]. 

Limiting  attention  to  purely  AutoRegressive  (AR)  based  vibration 
synthesis  leads  to  the  well-known  Yule-Walker  equations  which  are  linear  in 
terms  of  the  representation  parameters  [2].  Unfortunately  purely  AR  based 
schemes  are  ineffective  for  proper  synthesis,  especially  when  the  partially 
specified  covariance  corresponds  to  relatively  sharp  valleys  in  the  spectral 
domain  [3,4].  Purely  Moving  Average  (MA)  representations,  such  as  the 
ones  advocated  by  Abdul-Sada  and  Mahmood  [.5]  in  a  different  context, 
are  also  inappropriate  for  describing  the  relatively  sharp  spectral  peaks 


1534 


of  vibration  signals.  These  considerations  unambiguously  lead  to  mixed 
Autoregressive  Moving  Average  (ARM A)  based  vibration  signal  synthesis. 

Unfortunately,  ARM  A  l-)ased  synthesis  is  a  much  less  tractable  prob¬ 
lem,  leading  to  a  non-linear  mathematical  optimization  problem  [1-3].  To 
circumvent  this  difficulty  Gei'sch  and  Luo  [6]  proposed  a  mixed  scheme,  in 
which  the  AR  parameters  are  detained  through  the  Yule- Walker  equations, 
while  the  MA  parameters  are  obtained  via  a  non-linear  modified  Newton- 
Raphson  procedure.  In  subsequent  work,  Gei'sch  and  co-workers  [7,8]  uti¬ 
lized  proper  extensions  of  the  Two  Stage  Least  Squares  (2SLS)  method 
introduced  in  time  series  analysis  by  Durbin  [9].  The  2SLS  method  is,  nev¬ 
ertheless,  known  to  be  characterized  by  statistical  inefficiency  and  limited 
achievable  accuracy  [10].  In  related  work  within  a  broader  context,  Geor- 
gioii  [11]  attempted  finding  an  admissible  extension  of  a  partially  specified 
covariance  sequence  via  a  method  that  recursively  updates  ARMA  repre¬ 
sentations  of  dimension  increasing  with  the  sequence  length.  This  method, 
however,  requires  a-priori  information  on  the  representation’s  zero  loca¬ 
tions.  For  very  simple  cases,  corresponding  to  low  order  MA  polynomials, 
this  information  may  be,  perhaps,  derived  from  the  asymptotic  behavior 
of  the  partial  autocorrelation  coefficients.  Nevertheless,  such  a  procedure 
cannot  be  used  with  realistic  MA  polynomials  of  interest  in  the  vibration 
synthesis  problem. 

In  this  paper  an  effective  scheme  for  ARMA  based  vibration  signal 
synthesis  is  introduced.  The  scheme  is  based  upon  a  Fast  Rational  Model 
estimation  approach  [3,4],  combined  with  the  covariance  invariance  prin¬ 
ciple  and  a  dispersion  analysis  methodology.  The  covariance  invariance 
principle  [2]  is  used  for  relating  the  continuous  and  discrete  vibration  sig¬ 
nal  representations,  and  leads  to  a  special  ARMA(2n,2n-l)  structure  for  the 
latter.  The  Fast  Rational  Model  approach,  originally  introduced  within  the 
context  of  spectral  estimation  from  available  signal  samples  [3,4],  is  used 
for  effective  parameter  estimation,  while  the  dispersion  analysis  method¬ 
ology  [12]  is  used  as  the  main  tool  for  representation  order  determination. 
The  scheme  uses  the  specified  autocovariance  samples  as  a  pseudo-sufficient 
statistic,  and  develops  a  strongly  consistent  ARMA  representation  based 
upon  exclusively  linear  operations.  An  additional  important  property  of  the 
proposed  scheme  is  that  it  is  capable  of  mathematically  guaranteeing  the 
stability  and  invertibility  of  the  obtained  ARMA  representation;  a  feature 
implying  that,  unlike  alternative  techniques,  it  can  handle  the  practically 
significant  classes  of  vibration  signals  characterized  by  sharp  spectral  peaks 
or  valleys  without  any  difficulty. 

The  paper  is  organized  as  follows:  The  proposed  vibration  signal 
synthesis  methodology  is  presented  in  Section  2,  synthesis  test  cases  are 
considered  in  Section  3,  and  the  conclusions  of  the  work  are  summarized 
in  Section  4. 


2.  VIBRATION  SIGNAL  SYNTHESIS  METHODOLOGY 
2.1  Fundamentals 

Consider  a  linear  viscously-damped  rio  degree-of-freedom  structural 


1535 


system  described  Id.y  the  vector  differential  equation: 

M  •  x(t) -I- C  •  x(i) -f- K  ■  x(0  =  f{i)  (1) 

where  M,C,K  represent  the  Uo  x  Uo  mass,  viscous  damping,  and  stiffness 
matrices,  respective!}^,  and  f(^),  x(i)  the  ?7.o-dimensional  force  excitation 
and  vibration  displacement  signals,  respectively.  Laplace  transforming  (1) 
leads  to: 


X(s)  =  [M  •  .5-  -I-  C  •  s  -I-  K]-‘  •  F(.s)  =  G(s)  •  F(s)  (2) 

in  which  s  represents  the  Laplace  Transform  variable  and  G(s)  the  system’s 
receptance  transfer  matrix. 

Let  Gij[s)  represent  the  ij-th  element  of  G(s),  that  is  the  transfer 
function  relating  the  Lth  displacement  with  the  j-th  force.  Assuming  that 
the  force  excitation  is  a  stationary  zero-mean  imcorrelated  stochastic  signal, 
that  is: 


=  0  •  fjito)}  =  aj  ■  6{ti  -  £2) 


with  E{-}  indicating  statistical  expectation  and  5(-)  the  Dirac  delta  func¬ 
tion,  the  resulting  vibration  displacement  Xi{t)  will,  in  the  steady-state,  be 
a  stationary  stochastic  signal  with  zero  mean  and  autocovariance  [2]: 

roo 

7(r)=crP/  +  =  (3) 

k=l 

In  these  expressions  (the  latter  of  which  assumes  distinct  poles)  g{t)  repre¬ 
sents  the  impulse  response  of  the  receptance  transfer  function  Gij{s),  and 
dk  {k  =  1,2,  ••  •  ,2no)  the  structural  poles  and  corresponding  autoco¬ 
variance  expansion  coefficients,  respectively. 

The  uniform  instantaneous  sampling  of  the  stochastic  vibration  dis¬ 
placement  x{t)  (the  subscript  being,  for  the  sake  of  simplicity,  suppressed) 
leads  to  a  discrete-time  stochastic  signal  x[kT]  {k  =  0,1,2,-’*),  with  T  rep¬ 
resenting  the  sampling  period,  which  is  characterized  by  an  autocovariance 
function  'y[kT]  satisfying  the  covariance  invariance  principle  [2]: 

7[/:Tl=7('r)|r=/,-T  (4) 

Assuming  that  ,t(£)  is  band-limited,  and  that  the  conditions  of  the  Nyquist 
theorem  are  satisfied,  the  continuous  and  discrete  spectral  densities  satisfy 
the  relationship  [1]: 


S°(a.)  =  is7a,)  (_!<„<  1)  (5) 

in  which  oj  represents  frequency  in  rads /sec,  and  S^{u)  the  con¬ 

tinuous  and  discrete  densities,  respectively. 

Denoting  the  discrete  time  kT  in  normalized  form  as  simply  k,  or 
£  (£  =  0,1,2,---),  the  sampled  displacement  signal  may  be  shown  [2]  to 


1536 


admit  a  stable  and  invertible  AutoRegressive  Moving  Average  (ARM A) 
representation  of  orders  (2no,2?ro  —  1),  that  is: 


2710  2710-1 

x\t]  f  ^  (f)k  •  x[t  -  /c]  =  ry[tl  -!-  ^  Oj,  •  w[t  -  /c]  ==^ 

A:=l  k=l 

=>  •  x[t]  =  f){B)  ■  rult]  (6) 

In  the  above  iv\t]  represents  a  discrete,  zero-mean,  imcorrelated  (inno¬ 
vations)  sequence  with  variance  and  (f>ic  {k  =  1, 2,  ■  •  - ,  27io)  and  9k 
(/c  —  1,2,  •  •  • ,  271o  -  1)  the  autoregressive  (AR)  and  Moving  Average  (MA) 
parameters,  respectively.  The  AR  and  MA  polynomials  are  defined  as: 

<t>(B)  =  +  3{B)  = 

with  B  denoting  the  backshift  operator  (S  •  x[t]  =  a;[t  -  1]). 

Once  determined  from  the  specified  partial  covariance  sequence,  a 
proper  ARM  A  I'epresentation  of  the  form  (6)  may  be  used  for  vibration 
signal  synthesis. 

2.2  The  Algorithm 

The  vibration  synthesis  problem  treated  concerns  the  generation  (re¬ 
alization)  of  discrete  random  vibration  displacement  signals  with  covariance 
characteristics  conforming  to  a  prescribed  partial  (truncated)  autocovari¬ 
ance  sequence  ^[k]  (/c  ~  0, 1, 2,  •  -  ■ ,  i?). 

The  vibration  signals  of  interest  may  be  viewed  as  responses  of  linear 
viscously  damped  structural  systems  of  the  form  (1)  subject  to  uncorre¬ 
lated  force  excitation,  and  may  be  synthesized  via  discrete  ARMA(2n,2n-l) 
representations  of  the  form  (6)  driven  by  realizations  of  pseudo-random  un¬ 
correlated  sequences  characterized  by  zero  mean  and  variance  cr,^.  Toward 
this  end  iDoth  the  ARM  A  representation  and  innovations  varipce  cr^  need 
to  be  determined  from  the  given  covariance  information.  This  is  achieved 
via  the  Fast  Rational  Model  estimation  approach,  which  consists  of  a  se¬ 
quence  of  stages,  with  each  one  based  on  exclusively  linear  operations  [3,4], 
as  well  as  the  dispersion  analysis  methodology  (see  [12]  for  a  treatment 
of  the  notion  of  modal  dispersion)  for  representation  order  determination. 
The  resulting  synthesis  algorithm  may  be  outlined  as  follows: 

In  Stage  1  a  truncated,  p-th  order,  version  of  the  ARMA  represen¬ 
tation’s  inverse  function  {IjYj^i  {h  =  1)  [R  >  V  >  2?!  with  p  generally 
selected  -cis  p  ^  (2.5  -  5)  x  2n]  is  obtained  through  the  estimator  expres¬ 

sions; 

j2lj-l[k-j]+l[k]  =  0  (/c  =  l,2.--.,p)  (7) 


In  Stage  2  initial  MA  parameter  estimates  9°.  {k  =  1, 2,  •  •  • ,  272  —  1) 


1537 


are  obtained  by  solving  the  set  of  equations: 

2n-l 

=  (i  =  2n-|-l,---,4n-l)  (8) 

A:=l 

□ 

In  Stage  3  AR  parameter  estimates  d)]  (j  =  1, 2,  ■  ■  • ,  2n)  are  obtained 
via  the  estimatoj*  expressions: 

=  0  (/c  =  l,2,---.2n)  (9) 


in  which  the  superscript  i  denotes  iteration  number  and  7^  ^[fc]  the  auto¬ 
covariance  function  of  the  signal: 


1*1 


a:[t] 


(10) 


The  samples  of  this  autocovariance  are  computed  by  assuming 

7*"' [A]  ss  0  for  1*1  >  R.  This  leads  to  the  following  set  of  equations, 
which  may  be  solved  for  q'-'lfc]  (i:  =  0, 1,  •  •  • ,  R): 


X:A(i,;)-y-1i]  =  7b1 


1=0 


in  which: 


aj^i  I  =  0 

aj+i  -b  I  ^  0 


where  aj  =  0  for  j  0  (0,  2n  -  1],  and: 

2n~l~j 


k=0 


(11) 


(12) 


(13) 


□ 

In  Stage  4  updated  MA  parameter  estimates  0]-  (j  =  1, 2,  ■  •  • ,  2n  -  1) 
are  obtained  based  upon  the  expressions: 

9}  =  -  E  4  •  (i  =  1, 2,  •  •  ■ ,  2n  -  1)  (14) 


In  Stage  5  the  innovations  variance  is  estimated  as: 


(4)^  = 


T[0]  +  Ei*b'l'[l1 


j=l 


2n-l 

1-H 


n  -1 


(15) 


1538 


with  Cj  representing  an  estimate,  obtained  at  iteration  i,  of  the  j-th  sam¬ 
ple  of  the  AR.MA  representation’s  Green’s  function  [2].  This  is  achieved 
through  the  recursive  expressions: 

G;  =  «'-X;4-C%  (j=^l,2,---,2n-l)  (16) 

A:=l 


with  =  1,  G)  ^  0  for  j  <  0.  Following  this,  the  representation’s 
aiitoco variance  generating  function  and  spectral  density  may  be  obtained 
as: 


fiB)  ^  f:  7’ifcl-B"  = 

k=-oo 


0^{b)-0Hb-^)  o  , 


with  j  denoting  the  imaginary  unit. 


In  Stage  6  the  dispersion  percentage  <5^  [j  -  1, 2,  •  •  - ,  n)  of  each  vi¬ 
bration  mode,  expressing  the  mode’s  normalized  energy  contribution  in  the 
signal  representation,  is  (in  the  underdamped  case)  computed  as: 

=  (%)  {j  =  l,2,---,n)  (18) 

with  the  quantities  dj  (j  =  1, 2,  •  •  • ,  2?^)  being  defined  by  Eq.  (3).  □ 


Stages  1  through  6  are  repeated  for  signal  representations  with  suc¬ 
cessively  increasing  number  of  degrees  of  freedom  n,  and,  in  Stage  7,  the 
proper  number  of  degrees  of  freedom  is  selected  as: 

n*  =  max{  n  |  15j|  >  e  Vj}  (19) 

with  e  indicating  a  small  positive  threshold  value  selected  in  accordance 
with  the  desired  synthesis  accuracy.  O 

In  Stage  8  the  vibration  signal  is  synthesized  by  exciting  the  obtained 
ARMA(27^*,27^'‘  -  1)  representation  by  a  simulated  zero  mean  uncorrelated 
sequence  with  variance  ((7,^y)\  C 


Remark  1.  For  a  given  signal  representation  order,  stages  3  and  4  of  the 
algorithm  are  iterated  until  convergence  is  achieved.  The  estimation  results 
of  the  iteration  characterized  by  minimal  innovations  variance  (cr.^)^  are 
selected  as  best. 

Remark  2.  The  invertibility  of  an  estimated  ARMA{2n,2n-l)  representation 
may  be  mathematically  guaranteed  via  slightly  modified  MA  parameter  es¬ 
timators  in  stages  2  and  4,  that  are  based  upon  the  stability  property  of 
zero-lag  least-squares  inverses  (alternative  versions  A  and  B  in  Fassois  [4]). 
Alternative  version  B  may  be  used  in  stage  3  in  order  to  guarantee  the 
representation  stability  as  well.  The  guaranteed  stability  and  invertibility 
properties  are  of  primary  importance,  as  they  imply  that  all  types  of  vi¬ 
bration  signals,  including  those  characterized  by  sharp  spectral  peaks  or 
valleys,  may  be  synthesized  without  difficulty  (see  section  3). 


1539 


mi  =  0.5  kg 

ki  =  100  N/m 

Cl  =  0.5  N*sec/m 

m2  =  2.0  kg 

0 

0 

rH 

II 

C2  =  0.5  N*sec/m 

m3  =  1.0  kg 

*3  =  150  N/m 

C3  =  1.5  N*sec/m 

*4  =  100  N/m 

C4  =  1.5  N*sec/m 

cs  =  0.6  N*sec/m 
C6  =  0.5  N*sec/m 

Figure  1:  Schematic  representation  of  the  three  degree-of-freedom  struc¬ 
tural  system  of  Test  Case  I. 

3.  SYNTHESIS  TEST  CASES 

Two  test  cases,  in  which  the  scheme  is  used  for  the  synthesis  of  sig¬ 
nals  matching  a  specified  portion  of  the  covariance  structure  of  selected 
vibration  displacements,  are  presented. 

3.1  Test  Case  I 

In  Test  Case  I  the  synthesis  of  signals  matching  part  of  the  covari¬ 
ance  structure  of  the  displacement  of  mass  1  of  the  three  degree-of-freedom 
system  of  Figure  1  when  excited  at  that  point  by  a  continuous-time  uncor¬ 
related  force  is  considered.  This  type  of  signal  is  characterized  by  spectral 
peaks  at  1.04  and  3.32  Hz^  with  a  spectral  valley  between  them,  as  well  as 
a  much  less  obvious  third  peak  at  2.68  Hz  (solid  curve  in  Figure  2). 

The  auto  covariance  function  jlk]  is  specified  at  lags  k  ~  0, 1,2,  •  •  •, 
50  {R  =  50),  the  sampling  period  is  selected  as  T  =  0.075  secs,  the  inverse 
function  order  as  p  =  25,  and  the  maximum  allowable  number  of  iterations 
imax  —  25. 

The  spectral  density  of  the  synthesized,  theoretically  sufficient,  ARMA 
(6,5)  representation  is  compared  to  the  theoretical  spectral  density  in  Fig¬ 
ure  2(a),  from  which  very  good  agreement  is  observed.  A  similar  remark 
may  be  made  for  the  autocovariance  function  of  the  synthesized  represen¬ 
tation  (not  shown). 

Allowing  the  I'epresentation  order  to  be  increased  to  2n  =  8  [ARMA 


1540 


POWER  SPECTRUM  (dB) 


Figure  2:  Synthesized  ( - )  versus  theoretical  ( — )  spectral  density  func¬ 

tion:  (a)  ARM  A  (6,5)  based  synthesis,  (b)  ARM  A  (8,7)  based  synthesis 
(Test  Case  I;  fi  =  50). 


POWER  SPECTRUM  (dB) 


O 

z 


-1.0 


O  ~  SPECIFIED 
X  -  SYNTHESIZED 


1  0 
LAG 

(a) 


20 


(b) 


Figure  3;  ARM  A(  10,9)  based  synthesis  results:  (a)  synthesized  (x)  versus 
specified  (o)  auto  covariance  function,  (b)  synthesized  ( - )  versus  theoret¬ 

ical  ( — )  spectral  density  function  (Test  Case  1]  R  =  50). 


3  0 


TIME  (sec) 


Figure  4:  ARMA{10,9)  based  vibration  displacement  signal  realization 
(Test  Case  1;  R  =  50). 


(8,7)  model]  leads  to  visible  improvements  in  the  synthesized  representa¬ 
tion  spectral  density  [Figure  2(b)],  as  well  as  in  its  autocovariance.  These 
improvements  are  more  evident  at  autocovariance  lags  higher  than  10,  and 
at  frequencies  corresponding  to  the  neighborhood  of  the  spectral  valley 
located  between  the  two  peaks. 

A  further  increase  of  the  representation  order  to  2n  =  10  [ARMA(10,9) 
model]  leads  to  an  almost  perfect  match  of  both  the  specified  autocovari¬ 
ance  and  spectral  density,  as  the  theoretical  and  synthesized  curves  are 
practically  indistinguishable  (Figure  3).  The  extra  poles  introduced  in  this 
case  may  be  grouped  into  two  pairs:  a  real  and  a  complex  conjugate,  and  are 
both  characterized  by  small  dispersion  percentages.  These  results  suggest 
that,  despite  the  theoretical  sufficiency  of  the  ARMA(6,5)  representation, 
which  is  also  confirmed  by  the  Akaike  Information  Criterion  (AIC)  [1],  over¬ 
fitting  is  beneficial  in  achieving  the  highest  accuracy  in  matching  the  spec¬ 
ified  auto  covariance  characteristics.  Portion  of  a  vibration  displacement 
signal  realization  obtained  by  the  synthesized  ARMA(10,9)  representation 
is  depicted  in  Figure  4. 

The  effects  of  the  number  of  lags  R,  at  which  the  autocovariance 
function  is  specified,  have  been  also  investigated.  Lowering  R  to  35  (from 
its  current  value  of  50)  does  not  appreciably  change  the  synthesized  repre¬ 
sentation’s  spectral  density,  although  it  does  lead  to  slight  deterioration  in 
the  neighborhood  of  the  spectral  valley  and  at  the  high  frequency  end.  On 
the  other  hand,  increasing  R  to  100  produces  a  spectral  density  that  is  es¬ 
sentially  identical  to  that  of  the  R  =  50  case,  and  almost  indistinguishable 
from  the  theoretical. 

3.2  Test  Case  II 

In  Test  Case  II  the  synthesis  of  signals  matching  part  of  the  covari- 


1543 


mi  =  1,0 

kg 

II 

0 

0 

N/m 

m2  =  1-0 

kg 

0 

0 

II 

N/m 

m3  =  2.0 

kg 

0 

0 

II 

N/m 

0 

0 

r-t 

II 

►3? 

N/m 

Cl  =  0.6 

N*sec/in 

C2  =  0.5 

N*sec/m 

C3  =  0.6 

N*sec/m 

C4  =  1.5 

N*sec/m 

C5  =  0.5 

N*sec/m 

C6  =  0.7 

N*sec/m 

Figure  5:  Schematic  representation  of  the  three  degree-of-freedom  struc¬ 
tural  system  of  Test  Case  II. 


ance  structure  of  the  displacement  of  mass  1  of  the  three  degree-of-ffeedom 
system  of  Figure  5  when  excited  at  that  point  by  a  continuous- time  un¬ 
correlated  force  is  considered.  This  type  of  signal  is  characterized  by  three 
spectral  peaks  at  1.29,  2.03  and  2.81  Hz,  with  sharp  valleys  among  them 
[solid  curve  in  Figure  6(b)]  -  a  case  of  recognized  difficulty. 

The  aiitoco variance  function  'y[k]  is  specified  at  lags  k  —  0,1,2,  •••, 
50  {R  =  50),  the  sampling  period  is  selected  as  T  =  0.0884  secs,  the  inverse 
function  order  as  p  —  25,  and  the  maximum  allowable  number  of  iterations 
3-3  i-max  ~  25. 

Similarly  to  the  previous  case,  vibration  signal  synthesis  is  based  upon 
an  overfitted  ARM  A  (10, 9)  representation.  The  extra  poles  include  a  real 
and  a  complex  conjugate  pair,  and  are  all  characterized  by  small  dispersion 
percentages.  The  auto  covariance  and  spectral  density  of  the  synthesized 
ARMA(10,9)  representation  are  compared  to  those  specified  in  Figures  6(a) 
and  6(b),  respectively.  Evidently,  the  autocovariance  function  is  very  close 
to  the  specified,  and  certainly  very  satisfactory,  although  the  matching  is 
not  characterized  by  the  “perfection”  of  the  previous  case.  This  is  verified 
from  the  representation’s  spectral  density  plot  as  well,  which  indicates  some 
deviation  from  the  theoretical  density,  especially  in  the  neighborhood  of  the 
first  -  and  sharpest  -  spectral  valley.  This  is,  nevertheless,  expected,  due 
to  the  difficulties  associated  with  signals  characterized  by  sharp  spectral 
valleys.  The  fact  itself  that  the  algorithm  operated  properly  in  this  case  is 
due  to  its  important  inherent  stability  characteristics.  Portion  of  a  vibra¬ 
tion  displacement  signal  realization  obtained  the  estimated  ARMA(10,9) 


1544 


(a) 


(b) 


Figure  6:  ARM A(  10,9)  based  synthesis  results:  (a)  synthesized  (x)  versus 
specified  (o)  auto  covariance  function,  (b)  synthesized  ( - )  versus  theoret¬ 

ical  ( — )  spectral  density  function  (Test  Case  II;  R  =  50). 


Figure  7;  ARM  A  (10,9)  based  vibration  displacement  signal  realization 
(Test  Case  II:  R  —  50). 


representation  is  presented  in  Figure  7. 


4.  CONCLUDING  REMARKS 

In  this  paper  a  linear  multi-stage  scheme  for  effective  random  vibra¬ 
tion  signal  synthesis  from  partial  covariance  information  was  introduced. 
The  scheme  was  shown  to  achieve  accurate  synthesis  without  resorting  on 
nonlinear  operations.  Moreover,  by  providing  mathematically  guaranteed 
estimated  ARM  A  representation  stability  and  invertibility,  it  was  shown 
to  be  capable  of  effectively  synthesizing  all  types  of  vibration  signals,  in¬ 
cluding  the  difficult  classes  of  signals  characterized  by  sharp  spectral  peaks 
and/or  valleys.  In  particular: 


•  Signals  characterized  by  sharp  spectral  valleys  were  found  to  be  most 
difficult  to  synthesize,  with  the  synthesis  accuracy  generally  lagging 
in  the  neighl>orhood  of  the  spectral  valley. 

•  Representation  overfitting,  with  the  additional  modes  characterized 
by  relatively  small  dispersion  percentages,  was  shown  to  be  necessary 
in  achieving  maximum  synthesis  accuracy. 

•  The  required  minimal  number  of  autocovariance  samples  for  accurate 
synthesis  was  found  to  be  about  12no,  with  no  denoting  the  number  of 
structural  degrees  of  freedom.  Increasing  the  proportionality  factor 
to  about  16  was  shown  to  lead  to  some  improvement,  but  further 
increases  were  of  no  significance. 


1546 


REFERENCES 


[1]  SM.  Modern  Spectral  Estimation:  The.ory  and  Application.  Pre¬ 
ntice-Hall,  1988. 

[2]  Pandit  S.M.  and  Wu,  S.M.  Time  Series  and  System  Analysis  vjith 
Applications.  John  Wiley  and  Sons,  1983. 

[3]  Fassois,  S.D.  A  fast  rational  model  approach  to  parametric  spectral 
estimation.  Part  I:  the  algorithm.  ASME  Journal  of  Vibration  and 
Acoustics,  Vol.  112,  1990,  pp.  321-327. 

[4]  Fassois,  S.D.  A  fast  rational  model  approach  to  parametric  spectral 
estimation.  Part  II:  properties  and  performance  evaluation.  ASME 
Journal  of  Vibration  and  Acoustics,  Vol.  112,  1990,  pp.  328-336. 

[5]  Abdul-Sada,  J.W.  and  Mahmood,  M.K.  Generation  of  gaussian  pseudo¬ 
random  process  with  specific  correlation  properties.  International 
Journal  of  Systems  Science,  Vol.  19,  1988,  pp.  2163-2168. 

[6]  Gersch,  W.  and  Luo,  S.  Discrete  time  series  synthesis  of  randomly 
excited  structural  system  response.  Journal  of  the  Acoustical  Society 
of  America,  Vol.  51,  1972,  pp.  402-408. 

[7]  Gersch,  W.  and  Liu,  R.S-Z.  Time  series  methods  for  the  synthesis 
of  random  vibration  systems.  ASME  Journal  of  Applied  Mechanics, 
Vol.  98,  1976,  pp.  159-165. 

[8]  Gersch,  W.  and  Yonemoto,  J.  Synthesis  of  multivariate  random  vi¬ 
bration  systems:  a  two  stage  least  squares  ARMA  model  approach. 
Journal  of  Sound  and  Vibration,  Vol.  52,  1977,  pp.  553-565. 

[9]  Durbin,  J.  The  fitting  of  time  series  models.  Reviews  of  the  Interna¬ 
tional  Institute  of  Statistics,  Vol.  28,  1960,  pp.  233-244. 

[10]  Mayne,  D.Q.  and  Firoozan,  F.  Linear  identification  of  ARMA  pro¬ 
cesses.  Automatica,  Vol.  18,  1982,  pp.  461-466. 

[11]  Georgiou,  T.T.  Realization  of  power  spectra  from  partial  covariance 
sequences.  IEEE  Transactions  on  Acoustics,  Speech,  and  Signal  Pro¬ 
cessing,  Vol.  35,  1987,  pp.  438-449. 

[12]  Lee,  J.E.  and  Fassois,  S.D.  On  the  problem  of  stochastic  experi¬ 
mental  modal  analysis  based  on  multiple-excitation  multiple-response 
data.  Part  I:  dispersion  analysis  of  continuous-time  structural  sys¬ 
tems.  Journal  of  Sound  and  Vibration,  Vol.  161,  1993,  pp.  33-56. 


1547 


1548 


FIRST  PASSAGE  TIME  OF  MULTI-DEGREES  OF  FREEDOM 
NON-LINEAR  SYSTEMS  UNDER  NARROW-BAND 
NON-STATIONARY  RANDOM  EXCITATIONS 


C.W.S.  Tot  and  Z.  Ghent 

Department  of  Mechanical  Engineering 
University  of  Nebraska 
255  Walter  Scott  Engineering  Center 
Lincoln,  Nebraska  68588-0656 
U.S.A. 

E-mail:  cwsto@unlinfo.unl.edu 


Abstract 

The  study  of  motion  of  aerospace  systems  and  buildings,  that  house 
sophisticated  and  expensive  electronic  equipment  under  intensive  transient 
disturbances,  has  in  recent  years  become  an  important  issue  in  the  design  and 
analysis  process.  Central  to  the  study  is  the  problem  of  predicting  motion  and 
first  passage  time  of  the  system  under  such  disturbances. 

Owing  to  the  intensity  and  nature  of  the  excitation,  and  the  non¬ 
linearity  of  the  deformation  of  the  complex  system,  techniques  available  for 
the  response  predicting  is  very  limited.  To  provide  a  more  realistic  and 
accurate  prediction  of  response  and  assessment  of  the  first  passage  probability, 
in  this  paper  the  extended  stochastic  central  difference  method  is  proposed. 
The  recursive  response  statistics  and  first  passage  time  of  a  multi-degrees  of 
freedom  non-linear  system  under  excitations  treated  as  narrow  band  non¬ 
stationary  random  excitations  are  considered.  Results  of  a  two  degrees  of 
freedom  non-linear  system  indicate  that  the  technique  proposed  is  very 
efficient  and  accurate  compared  with  the  Monte  Carlo  simulation  data. 


Professor  and  corresponding  author 

Research  Assistant,  Department  of  Mechanical  Engineering,  The  University  of 
Western  Ontario,  London,  Ontario,  Canada  N6A  5B9 


1549 


1.  INTRODUCTION 


The  study  of  motion  of  aerospace  systems  and  buildings,  that  house 
sophisticated  and  expensive  electronic  equipment  under  intensive  transient 
disturbances,  has  in  recent  years  become  an  important  issue  in  the  design  and 
analysis  process.  Central  to  the  study  is  the  problem  of  predicting  motion  and 
first  passage  time  of  the  system  under  such  disturbances. 

Owing  to  the  intensity  and  nature  of  the  excitation,  modulated  wide 
band  random  excitation  process  is  a  highly  idealization  and  consequently,  even 
for  a  two  degrees  of  freedom  (dof)  linear  system  it  can  lead  to  large 
differences  between  the  responses  of  the  system  under  such  an  excitation 
process  and  those  using  a  more  realistic  representation  of  narrow  band  non¬ 
stationary  random  disturbance  [1].  For  a  single  dof  nonlinear  system,  the  first 
passage  time  based  on  wide  band  random  excitation  is  very  significantly 
different  from  that  using  the  narrow-band  random  excitation  process  [2].  A 
survey  of  the  literature  seems  to  suggest  that  no  analytical  or  computational 
technique,  bar  the  Monte  Carlo  simulation  (MCS)  method,  is  available  for  the 
prediction  of  motion  and  first  passage  time  of  multi-degrees  of  freedom 
(mdof)  nonlinear  systems  under  narrow  band  non-stationary  random  excitation. 
When  the  number  of  dof  is  large  and  the  excitation  is  a  narrow  band  non- 
stationary  process,  even  the  application  of  the  MCS  can  be  difficult  if  not 
impossible.  The  difficulty  lies  in  the  choice  of  centre  frequency,  bandwidth 
and  amplitude  of  the  response  from  the  filter  in  order  to  model  the  narrow 
band  process  that  is  being  approximated  and  applied  to  the  non-linear  system. 
To  provide  a  more  realistic  and  accurate  prediction  of  response  and  assessment 
of  the  first  passage  probability,  in  this  paper  the  extended  stochastic  central 
difference  (SCD)  method  is  proposed.  The  recursive  response  statistics  and 
first  passage  time  of  the  mdof  non-linnear  system  under  excitations  treated  as 
narrow  band  non-stationary  random  excitations  are  considered. 

The  organization  of  this  paper  is  as  follows;  a  brief  introduction  to  the 
extended  SCD  method  is  given  in  the  next  section  which  is  followed  by  a 
consideration  of  the  two  dof  system  under  narrow  band  non-stationary  random 
excitation  in  Section  3.  The  versatile  statistical  linearization  (SL)  or  equivalent 
linearization  (EL)  technique  is  applied  at  every  time  step  to  linearize  the  two 
dof  non-linear  system.  The  scheme  adopted'liere  is  scheme  TV  in  reference  [3]. 
Section  4  is  concerned  with  the  modified  adaptive  time  scheme  (ATS)  and  the 
first  passage  probabilities.  Computed  results  are  included  in  Section  5  while 
remarks  are  in  Section  6. 


1550 


2,  STOCHASTIC  CENTRAL  DIFFERENCE  METHOD  FOR  NARROW-BAND 
RANDOM  EXCITATIONS 

Consider  a  mdof  system  under  narrow  band  random  excitations,  which 
are  obtained  from  output  of  a  filter  perturbed  by  modulated  Gaussian  white 
noise  excitations.  The  governing  equations  of  motion  in  matrix  notation  are 

^ff  +  ^ff  +  ^ff  =  >  ( I  a,b) 


where  Mf,  Cf  and  Kf  are  the  mass,  damping  and  stiffness  matrices  of  the  filter 
while  My,  Cy  and  Ky  are  the  mass,  damping  and  stiffness  matrices  of  the 
system  respectively;  f  is  the  stochastic  displacement  vector  of  the  filter,  and 
Y  is  the  stochastic  displacement  vector  of  the  system;  e(t)  is  a  vector  of  time- 
dependent  deterministic  modulating  functions;  the  over-dot  and  double  over 
dot  are  the  first  and  second  derivatives,  respectively;  w(t)  is  a  zero  mean 
Gaussian  white  noise  process. 

Recursive  expressions  for  the  mdof  filter  and  mdof  system  described 
by  equation  (1)  can  be  derived  [1].  For  example,  the  recursive  expression  for 
system  responses  can  be  shown  as 

(A0‘‘N„K/s)<  +  N^yDyis)Nl  *  N,yDj(s)Nl  ^2) 

+  (At)'Nj^G(s)W,;  +  (AtfN,yG(s)'^Nl 
*  iAtfN,yH(.s)Nl  * 

where  R,(s)  =  Ry(s)  =  <Y,Y7>;  D  (s)  =  <Y,Y,.,'">  =  R,(s-1)  +  N,, 

D/Cs-l)  +  (At)'  Ni,  G(s-l)'',  G(s)  =  <Y,f7>,  H(s)  =  <Y„,f7>,  f,  =  f(s)  is  f  at 
time  step  t„  At=t„,-ts.  Y.  =  Y(s)  is  Y  at  time  step  t„  the  angular  brackets 
denote  ensemble  average  of  the  enclosing  quantity  while  the  superscript  T 
desingates  the  transpose  of  the  matrix; 

N,y  -IMy*  (Atm  Cy]-^ ,  '  N^y  [  2M^  "  (AO^iT^] , 

N^y-N,yl{Atl2)Cy-My],  - 
G(s)  -  (AtfN^yD^(s)  *  N^yH(s) 

+  N,yH{s-l)  *  N,y  G(s-2) 

H(s)  -  GCs-DW,^*  (.AtfN^yRp-2)N,^f 
*  N^y  G{s-2)  Ny  *  N,yH(s-'i)  K- 


1551 


Equation  (2)  is  applicable  to  systems  under  multiple  narrow  band  random 
excitations  generated  by  the  filter  matrix  equation. 

In  order  to  provide  the  required  response  statistics  for  the  first  passage 
probability  computation,  the  covariance  of  displacement  and  velocity 
responses,  and  the  covariance  of  velocity  response  are  considered  in  the 
following.  Starting  from  the  central  difference  method,  the  velocity  vector  at 
the  current  time  step  is  given  by 

f(s)  .  ^[y(y+l)-7(s-l)] 

2Ar 

which  upon  post  multiplying  the  transpose  of  displacement  vector  Y(s)  gives 
the  covariance  matrix  of  velocity  and  displacement  as 

( r(^)  r '■(s) )  -  ^  [  D(s*  1)  -  D  ^(s)  ] ,  (3) 

since  D'''y(s)  =  =  <Y5.,YJ>.  Similarly,  the  covariance  matrix  of 

velocity  responses  becomes 

f  5(s+l)  +  fe-l)  -  (AtfN  H^(s) 

-  N^,D^(s)  -  N,^R^(s-D  (4) 

The  foregoing  is  the  so-called  extended  SCD  method.  When  the  excitations  are 
wide  band  stationary  or  non-stationary  random  processes  the  above  equations 
reduce  to  those  of  the  SCD  method.  Equations  (2),  (3)  and  (4)  can  be  applied 
to  the  response  analysis  of  mdof  systems  under  narrow  band  non-stationary 
random  excitations.  The  corresponding  first  passage  probability  can  be 
evaluated  as  in  reference  [4].  For  non-linear  systems  whose  nonlinearities  are 
explicitly  defined,  however,  the  following  technique  is  employed  in 
conjunction  with  the  modified  ATS  before  the  computation  of  first  passage 
probabilities  can  be  performed. 


3.  NON-LINEAR  SYSTEMS  AND  STATISTICAL  LINEARIZATION 

Consider  the  two  dof  system  shown  in  Figure  1.  The  system  is 
subjected  to  a  narrow  band  non-stationary  random  excitation  at  its  base  and 
the  restoring  force  of  the  spring  connecting  the  two  masses  Mj  and  M2  has 
quadratic  and  cubic  nonlinear  terms  associated  with,  respectively,  parameters 


1552 


T]’  and  e’.  This  two  dof  system  under  a  modulated  white  noise  excitation  was 
considered  by  Kimura  and  Sakata  [5],  and  Liu  and  To  [3,  6].  Practical 
examples  that  can  be  modeled  by  this  system  are;  (a)  soil-structure  coupled 
system,  (b)  a  primary  building  structure  with  secondary  system  representing 
installed  equipment  under  an  earthquake  excitation,  and  (c)  an  offshore  oil 
platform  with  a  secondary  equipment  such  as  the  drill  string  tower  under 
strong  wave  impact. 

If  one  introduces  relative  displacements  or  inter-story  drift  in  the 
language  of  earthquake  engineering  such  that  =  yi  -  yo  and  Y2  =  y2  -  yi, 
where  yg,  yi  and  are,  respectively,  the  absolute  displacements  with  respect 
to  the  base,  the  matrix  equation  of  motion  can  be  expressed  as  [3,  5,  6] 


1  o' 

jM 

>  + 

^—4 

0^ 

2C,W  ■ 

2(1  +  p)C2 

I  (xtiy/  - 


r-' 

«  + 

ri 

i'. , 

-W^  1+li 

In’l 


(5) 


or  written  in  a  more  compact  form 

M  Y  ^  C  Y  ^  K^Y  ^  g{Y)  =  r(T) 


(6) 


in  which 


1  o' 

\2i;,w 
\  ,  c  =  \ 

0  1, 

1  [-2C,W 

2(1+vi)C2 

-JV^ 


-P 

Up 


g(X)  - 


(h-p)Ti72^  +  a  + 


Y  = 


r(T)  = 


e(T)v/7| 


1553 


where  the  over-dot  and  double  over-dot  denote,  respectively,  the  first  and 
second  derivatives  with  respect  to  T.  In  equation  (6)  the  non-stationary  random 
excitation  is  represented  as  a  product  of  a  deterministic  amplitude  modulating 
function  e(T)  and  a  zero  mean  narrow  band  random  process  which  is  related 
to  the  response  from  the  filter.  The  symbol  I  is  assumed  to  be  constant.  Note 
that  the  following  definitions  have  been  applied  in  equations  (5)  and  (6) 

(0^  -  ,  <4  -  ,  2CiQi  -  Cj/Mj  , 


-  Cj/Mj  .  Ti  -  . 


e  -  ,  V-  '  , 


such  that  equations  (5)  and  (6)  are  dimensionless.  Therefore,  the  symbol  T  in 
this  section  should  not  be  confused  with  the  T  used  in  the  time  co-ordinate 
transformation  (TCT)  in  reference  [6]. 

The  discretization  of  equation  (6)  in  the  x  domain  leads  to 

M  f,  +  C  f,  +  Jfo  r,  +  g(y,)  -  r,  (8) 

where  the  subscript  s  is  a  positive  integer  denoting  the  time  step  such  that 
Ax  =  X3+1  -Xj  and  Xq  =  0. 

Since  this  is  a  non-linear  system,  the  mean  of  its  response  will  in 
general  be  non-zero  even  though  the  excitation  is  of  zero  mean.  It  is  generally 
agreed  that  the  assumption  of  Gaussian  response  in  the  linearization  technique 
of  Caughey  [7]  is  approximately  satisfied  for  weakly  nonlinear  systems.  In  the 
present  investigation  the  highly  nonlinear  systems  are  approximated  as  a  series 
of  weakly  nonlinear  or  linear  systems  and  therefore  one  can  assume  that  the 
response  at  every  time  step  is  Gaussian.  Experience  [3,6]  with  nonlinear 
systems  under  wideband  random  excitations  shows  that  such  an  assumption  is 
acceptable  as  very  accurate  results  were  obtained.  In  spirit  similar  to 
references  [3,6],  equation  (8)  can  thus  be  represented  by  the  following 
linearized  equation 


1554 


where  K^q  is  the  equivalent  stiffness  matrix  which  is  time  dependent.  It  is 
determined  by  the  following  equation 


BgfJis)) 


ij  =1,2. 


Upon  substituting  the  non-linear  term  g(Y)  into  equation  (10)  and  carrying  out 
the  operation  one  has 

0  2\ir\<Y^is)>  -  3p.t<Y2is)> 

K(s)  -  K.  +  ,  •  ^  ^ 

0  2(l-p)Ti<y2(5)>  +  3{U\i)B<Y^is)> 


With  equation  (11),  one  can  then  apply  the  SCD  technique  to  equation  (9)  and 
perform  the  operation  to  obtain  the  recursive  expression  which  is  identical  to 
that  described  in  Section  2  above.  The  modified  ATS  to  be  introduced  in  the 
next  section  has  to  be  employed  with  the  SCD  method  to  update  the  K^q  at 
every  time  step. 

As  the  system  is  non-linear  and  therefore  ensemble  averages  of 
responses  are  not  zero  in  general,  however,  the  following  recursive  ensemble 
average  of  response  vector  clearly  indicate  that  it  is  zero  if  the  system  starts 
from  rest 

m(5+l)  =  N2y(s)m(s)  +  N^ym(s-l)  .  (12) 

where  m(s+l)  is  the  ensemble  average  of  system  response  vector  at  the  next 
time  step. 

To  circumvent  this  problem,  the  g(Y)  term  of  equation  (6)  is  moved 
to  the  right  hand  side  (RHS)  such  that 

M  ns)  *  C  y(s)  +  r(s)  -  r(s)  -  g(r(i))  .  (13) 

As  will  be  seen  in  the  following  this  equation  is  applied  only  for  the  initial 
states  of  the  response  predictions.  Substituting  the  central  difference 
approximation  of  velocity  and  acceleration Jerms 

y(s)  -  — [  y(s+i)  -  y(s-i)  ] , 

2  At 


y(i+i)  -  2  ns)  *  i'(s-i)] . 


(14) 


(15) 


into  equation  (13)  one  has 

y(i+l)  -  N^^Y(s)  +  JVj, 7(5-1)  + 
-  (AT)2Wi^g(y(5)) , 


where 


Nj,  =  [  2M  -  iAx)%  ]  . 


(16) 


Taking  the  ensemble  average  of  equation  (15)  one  obtains 
mis^l)  -  mis)  +  m(5-l)  -  {Ax)^  N^y<giYis))>  , 


<g(Yis))> 


^nl<y2(5)^>  - 

n  e  [  3  <y,(5)><yj(s)^>  -  2  <y2(5)>^  ] 


(i-ti)Ti<yj(s)">  f 

(i+n)e[3<y2(5)><y2(5)2>-2<yj(5)>^] 


(17) 


Substituting  equation  (17)  into  equations  (2)  and  (13)  one  can  show 

that 

R  (2)  .  [Ax)^N,^RJiDN^/  ,  m(2)  -  [0  0]^, 

(l6) 

m(3)  -  -(AT)S<l'2W^>iV,,[p  !-(»]’■• 

As  soon  as  the  above  non-zero  ensemble  averages  are  found,  one  can  return 
to  equation  (12).  For  s  >  3,  equation  (12)  can  be  applied  to  obtain  the  non¬ 
zero  ensemble  averages.  Of  course,  the  recursive  mean  squares  are  given  by 
equation  (2).  Up  to  this  stage,  the  SCO  method  with  modified  ATS  technique 
can  be  applied  to  the  system  to  compute  its  responses  to  random  excitations. 


4.  MODIFIED  ADAPTIVE  TIME  SCHEME  AND  FIRST  PASSAGE  PROBABILITIES 

As  described  by  Liu  and  To  [6],  the  time  step  size  in  a  non-linear 
system  varies  with  the  variance  of  displacement  response,  the  time  step  size 
has  to  be  updated  accordingly.  This  strategy  is  known  as  the  adaptive  time 
scheme  (ATS).  In  the  present  analysis  the  terms  that  have  to  be  considered  for 


1556 


the  time  step  updating  are  Ry(s-l),  G(s-2),  H(s-2),  Rf(s-2),  Niy,  N2y  and  N3y. 
This  has  become  more  critical  in  the  extended  SCD  method  because  the  error 
would  come  from  not  only  the  system  but  also  the  responses  of  the  filter.  The 
interpolation  scheme  is  employed  to  compute  the  time  step  size  when  it  is 
warranted  to  do  so  for  the  nonlinear  system  whose  natural  frequency  at  every 
time  step  is  different. 

With  the  variances  of  displacement  and  velocity  at  every  time  step 
computed  by  the  extended  SCD  method,  the  first  passage  problem  can  now  be 
considered.  Approximate  first  passage  probabilities  based  on  the  modified 
mean  rate  of  various  crossings  proposed  earlier  by  the  first  author  [4]  are 
computed.  In  the  latter  the  trapezoidal  rule  was  employed  to  evaluate  the  first 
passage  probabilities  and  uniform  time  step  was  assumed.  For  nonlinear 
system  employing  the  SCD  or  extended  SCD  method  and  modified  ATS,  the 
time  steps  are  different.  Therefore,  the  interpolation  scheme  is  applied  in  the 
present  investigation  such  that  the  trapezoidal  rule  can  be  adopted  for  the 
computation  of  first  passage  probabilities. 


5.  NUMERICAL  RESULTS 

Numerical  examples  are  presented  in  the  following.  Parameters  of  the 
system  in  Section  3  are: 

IF  =  p  =  1.0  ,  Cl  =  ^2  = 

such  that  the  two  dimensionless  natural  frequencies  of  the  corresponding  linear 
system,  that  is  when  T\  and  s  are  both  equal  to  zero,  are  cOji  =  0.618  and  (£>,2 
=  1.618.  The  amplitude  modulating  function  e(T)  chosen  is 

e(T)  -  -  e-o")  .  ^20) 

Numerical  results  presented  in  this  section  include  two  examples.  The 
first  example  is  the  one  with  the  centre  frequency  of  the  one  dof  cOf  =  1.0  and 
the  second  is  with  cOf  =  1.618.  Each  example  has  a  two  dof  system  described 
in  Section  3  and  a  single  dof  filter.  The  latter  equals  to  one  of  the  two  linear 
dimensionless  natural  frequencies  of  the,  non-linear  system.  The  other  is 
between  the  two  linear  dimensionless  natur^  frequencies.  Damping  ratio  of  the 
filter  (band  width)  in  the  two  examples  are  =  0.01,  0.1  and  1.0.  The  narrow 
band  random  excitation  to  the  system  is  the  response  taken  directly  from  the 
filter  which  is  perturbed  by  a  non-stationary  zero  mean  Gaussian  white  noise. 
The  spectral  density  of  the  Gaussian  white  noise  is  Sq  =  0.00012.  The  narrow 
band  random  excitation  to  the  system  is  at  its  base.  MCS  is  also  carried  out 
in  order  to  verify  the  results  by  the  extended  SCD  method.  Other  pertinent 
system  parameters  are  t]  =  -1.0  and  e  =  1.5. 


1557 


It  was  observed  that  the  results  of  the  case  where  cOf  =  1.0  and  = 
0.01  became  unstable  for  both  the  extended  SCD  and  MCS  methods.  An 
attempt  was  also  made  to  compute  responses  of  the  system  excited  by  a 
narrow  band  process  with  cOf  =  0.618.  In  this  case,  instability  occurs  when 
equals  to  0.01  and  0.1.  Note  that  the  spectral  density  of  the  white  noise 
process  is  one  order  of  magnitude  smaller  than  what  was  used  in  reference  [6]. 
This  is  because  when  Sq  =  0.0012  was  used,  the  response  of  the  system 
becomes  unstable.  Representative  response  statistics  for  Sq  =  0.0012  are 
present  in  Figures  2  and  3  in  which  the  covariances  of  displacement  and 
velocity  responses  were  not  included  for  clarity  since  they  are  very  close  to 
the  variance  <Y2>-  From  the  figures,  one  observes  that  the  results  by  the  SCD 
method  has  an  excellent  agreement  with  those  using  the  MCS.  The 
computational  time  using  a  Silicon  Graphics  Inc.  engineering  workstation  with 
64  megabyte  random  access  memory  and  60  mega  Hz  single  central  processor 
the  extended  SCD  method  is  approximately  1 5  seconds  while  that  for  the  MCS 
is  about  57  minutes.  Consequently,  one  can  conclude  that  the  extended  SCD 
method  is  very  efficient  and  accurate  compared  with  the  MCS  data. 

For  the  first  passage  probabilities,  two  sets  of  results  were  studied. 
They  are:  (i)  results  concerned  with  narrow  band  non-stationary  random 
excitations,  and  (ii)  comparison  of  narrow  band  non-stationary  random 
responses  to  wide  band  non-stationary  random  responses.  It  was  observed  that 
in  term  of  first  passage  probability  the  difference  between  narrow  band  non- 
stationary  and  narrow  band  stationary  random  excitations  is  insignificant  and 
therefore  these  results  are  not  presented  here.  In  set  (i),  results  obtained  by  the 
MCS  technique  are  included  for  direct  comparison.  Some  typical  results  of 
first  passage  probability  Ld  based  on  the  two  stage  process  with  allowance  for 
the  actual  duration  of  clumps  at  low  threshold  levels  are  presented  in  Figures 
4  and  5.  The  results  for  set  (ii)  are  plotted  in  Figures  6  and  7.  In  these  figures 
NB  designates  narrow  band,  and  WB  wide  band.  With  reference  to  the  results 
in  Figure  7,  one  can  conclude  that  the  first  passage  probability  of  the  model 
with  narrow  band  random  excitation  is  very  much  different  from  that  with 
wide  band  random  excitation. 

Before  leaving  this  section  it  may  be  appropriate  to  note  that  although 
the  filter  and  system  parameters  for  sets  (i)  and  (ii)  are  identical  the 
corresponding  plots  in  Figures  5  and  7  are  different.  For  the  results  in  set  (ii) 
and  plots  in  Figures  6  and  7  the  input  ^  the  system  is  a  product  of  an 
envelope  function  and  a  stationary  narrowband  random  process.  However,  for 
the  results  in  set  (i)  and  plots  in  Figures  4  and  5  the  input  to  the  system  is  the 
output  of  the  filter  whose  input  is  a  product  of  an  envelope  function  and  a 
zero  mean  stationary  white  noise  process.  In  other  words,  in  the  MCS  one  is 
unable  to  have  an  absolute  control  over  the  exact  narrowband  nonstationary 
input  to  the  system.  The  extended  SCD  method,  however,  has  no  such  a 
restriction  and  therefore  has  an  added  advantage  over  the  MCS  in  controlling 
the  input  to  the  system. 


1558 


6.  REMARKS 


In  this  paper  the  extended  stochastic  central  difference  method  is 
introduced.  The  recursive  response  statistics  and  first  passage  time  of  a  multi¬ 
degrees  of  freedom  non-linear  system  under  excitations  treated  as  narrow  band 
non-stationary  random  excitations  are  considered.  Results  of  a  two  degrees  of 
freedom  non-linear  system  indicate  that  the  technique  proposed  is  very 
efficient  and  accurate  compared  with  the  Monte  Carlo  simulation  data. 

One  also  observes  that  the  first  passage  probability  of  the  two  dof 
model  under  narrow  band  random  excitation  is  very  much  different  from  that 
with  wide  band  random  excitation.  This  suggests  that  correct  representation  of 
the  excitation  process  as  a  wide  band  or  narrow  band  random  process  is 
extremely  important  if  more  reliable  conclusions  are  to  be  drawn  and  used  in 
the  design  process. 


ACKNOWLEDGEMENT 

The  investigation  reported  above  was  supported  in  the  form  of  a 
research  grant  by  the  Natural  Sciences  and  Engineering  Research  Council  of 
Canada  awarded  to  the  first  author. 


REFERENCES 

1.  To,  C.W.S.  and  Chen,  Z.,  Response  analysis  of  system  under 
narrow  band  stationary  and  nonstationary  random  excitations.  In  Proceedings 
of  A.S.M.E.  Pressure  Vessels  and  Piping  Conference,  Montreal,  Canada,  July 
21-26,  1996,  PVP-Vol.  331,  pp.  107-114. 

2.  To,  C.W.S.  and  Chen,  Z.,  First  passage  time  of  nonlinear  ship 
rolling  in  nonstationary  random  Seas.  In  Proceedings  ofA.S.C.E.  7th  Specialty 
Conference  on  Probabilistic  Mechanics  and  Structural  Reliability,  Worcester, 
Massachusetts,  U.S.A.,  August  7-9,  1996,  pp.  250-253. 

3.  To,  C.W.S.  and  Liu,  M.L.,  Recursive  expressions  for  time 
dependent  means  and  mean  square  responses  of  a  multi-degree-of-freedom 
nonlinear  system.  Comput.  Struct.,  1993,  4^  (No.  6),  993-1000. 

4.  To,  C.W.S.,  Distribution  of  the  first  passage  time  of  mast  antenna 
structures  to  nonstationary  random  excitation.  /.  Sound  and  Vibration,  1986, 
108,  11-23. 

5.  Kimura,  K.  and  Sakata,  M.,  Nonstationary  response  analysis  of  a 
nonsymmetric  nonlinear  multi-degree-of-freedom  system  to  nonwhite  random 
excitation.  JSME  Int.  J.,  1988,  31(4),  690-697. 


1559 


6.  Liu,  M.L.  and  To,  C.W.S.,  Adaptive  time  schemes  for  responses  of 
nonlinear  multi-degree-of-freedom  systems  under  random  excitations.  Comput. 
Struct.,  1994,  52(No.  3),  563-571. 

7.  Caughey,  T.K.,  Equivalent  linearization  techniques.  J.  Acoust.  Soc. 
Am.,  1963,  35,  1706-1711. 


V. 


Figure  1.  The  two  dof  non-symmetric  non-linear  system 


1560 


1564 


RANDOM  RESPONSE  OF  DUFFING  OSCILLATOR 
EXCITED  BY  QUADRATIC  POLYNOMIAL  OF 
FILTERED  GAUSSIAN  NOISE 


C.  Floris  and  M.C.  Sandrelli 

Department  of  Structural  Engineering,  Politecnico  di  Milano,  Piazza 
Leonardo  da  Vinci  32, 1-20133  Milano,  Italy 


ABSTRACT 

The  random  response  of  a  Duffing  oscillator  excited  by  a  quadratic 
polynomial  of  filtered  Gaussian  process  is  considered.  The  problem  is 
approached  by  the  use  of  Ito's  stochastic  differential  calculus.  Being 
the  system  nonlinear,  the  moment  equations  (ME)  constitute  an  infinite 
hierarchy,  to  close  that  a  procedure  different  from  the  classical  cumu- 
lant-neglect  closure  method  is  used.  This  procedure  operates  in  two 
phases,  in  the  former  of  which  the  system  is  linearized  so  that  the  ME 
are  solved  exactly.  In  the  second  phase  the  actual  system  is  considered: 
the  equations  for  the  moments  of  m-th  order  contain  moments  of  order 
m+2.  To  solve  them  an  iterative  scheme  is  used,  in  the  first  step  of 
which  the  higher  order  moments  are  considered  as  known  quantities 
and  take  the  values  that  have  been  estimated  on  the  linearized  system. 
In  the  applications  the  new  procedure  is  shown  to  converge  towards 
the  solution  obtained  by  simulation  requiring  a  charge  for  calculation 
considerably  lesser  than  that  of  cumulant-neglect  closure  method.  Care 
is  taken  to  detect  possible  multiple  solutions  when  the  excitation  is 
narrow-banded. 


1.  INTRODUCTION 

The  study  of  nonlinear  dynamical  systems  under  random  excitation 
of  polynomial  form  is  of  considerable  interest  in  the  theory  of  random 
vibration  as  well  as  in  engineering  applications.  In  fact,  important  ran¬ 
dom  dynamical  agencies  can  be  properly  represented  as  polynomial 
forms  of  filtered  Gaussian  processes.  This  is  the  case  of  gusty  wind  [1] 
and  random  sea  waves  [2-4].  In  this  way,  both  the  system  and  the  exci¬ 
tation  are  nonlinear,  which  increases  the  difficulties  in  characterizing 
the  response  statistically.  System  response  is  probably  well  far  from 
Gaussianity,  since  the  non-normality  is  caused  by  both  the  non- 


1565 


normality  of  the  input  (memoryless  transformation  of  normal  process) 
and  non-linearity  of  the  response  process  itself. 

Nonlinear  random  vibration  problems  can  be  approached  by  sever¬ 
al  ways,  such  as  equivalent  linearization,  perturbation,  quasi-harmonic 
method,  multiple  time  scales,  and  Ito's  stochastic  differential  calculus. 
In  this  paper  the  last  approach  is  followed.  It  is  possible  since  the  exci¬ 
tation  is  idealized  as  the  output  of  a  linear  filter  excited  by  a  Gaussian 
white  noise.  Ito's  stochastic  calculus  provides  a  straightforward  proce¬ 
dure  for  deriving  the  differential  equations  in  terms  of  response  mom¬ 
ents  or  in  terms  of  the  probability  density  function  [5-9].  Nevertheless, 
in  presence  of  nonlinearities  in  both  the  excitation  and  the  system  man- 
y  difficulties  arise.  The  so-called  Fokker-  Planck-Kolmogorov  (FPK)  e- 
quation,  which  gives  the  probability  density  function  of  response,  can 
be  solved  analitically  in  very  few  cases,  especially  when  the  transient 
response  is  considered  (e.g.  see  [10]).  If  the  moment  equation  (ME)  ap¬ 
proach  is  used,  the  equations  of  moments  of  order  k  involve  higher  and 
lower  order  moments  than  k,  that  is  the  ME  constitute  an  infinite  hier¬ 
archy.  In  order  to  solve  this  problem  closure  schemes  have  been  pro¬ 
posed  [9,11,12].  Among  the  closure  schemes  the  cumulant-neglect  clo¬ 
sure  method  is  certainly  the  most  popular.  However,  although  this 
method  is  generally  efficacious,  it  has  some  shortcomings.  The 
moments  of  order  higher  than  k  that  appear  in  the  equations  for  the 
moments  of  order  k  are  expressed  as  a  function  of  the  lower  order 
moments  setting  the  corresponding  cumulants  equal  to  zero.  Since 
these  functions  are  nonlinear,  the  ME  too  become  nonlinear.  In  such  a 
way  computational  difficulties  arise.  Moreover,  a  nonlinear  system 
may  have  more  acceptable  solutions  (it  is  recalled  that  a  solution  is 
acceptable  when  ail  even  moments  are  positive  and  Cauchy-Schwartz 
inequality  is  satisfied).  These  multiple  solutions  must  be  considered 
with  care,  possibly  determining  whether  they  are  stable  or  not.  The 
significance  of  multiple  solutions  in  terms  of  statistical  moments  of  the 
response  of  a  nonlinear  dynamical  system  is  still  an  open  question  [IS¬ 
IS].  However,  in  some  cases  such  as  a  Duffing  oscillator  multivalued 
response  moments  are  expected  in  the  presence  of  a  narrow-band 
excitation  [17].  These  different  values  of  variance  correspond  to  differ¬ 
ent  local  states  around  which  the  system  oscillates  with  abrupt  jumps 
between  the  different  states.  Furthermore,  the  steady-state  probability 
density  function  (PDF)  of  response  is  bimodal  or  multimodal,  even  if  it 
is  unique,  since  FPK  equation  has  one  solution  only.  In  other  cases  that 
have  been  studied  by  the  writers  and  Prof.  Di  Paola  [18]  the  simulated 
response  does  not  show  any  jump  and  the  significance  of  multivalued 
response  is  not  clear. 


1566 


In  this  paper  a  new  closure  scheme  is  adopted.  The  procedure  oper¬ 
ates  in  two  phases,  in  the  former  of  which  advantage  is  taken  of  the 
fact  that  the  ME  of  linear  systems  excited  by  polynomial  forms  of 
Gaussian  processes  [2,  19]  or  delta-correlated  processes  [20,21]  are  ex¬ 
actly  solvable.  Hence,  the  nonlinear  system  is  replaced  by  a  linear  one, 
whose  parameters  are  chosen  in  such  a  way  to  minimize  the  error 
between  the  original  and  the  linearized  system  in  some  statistical  sense 
[22,23].  The  statistical  moments,  which  are  exact  for  the  latter,  are  cal¬ 
culated  to  an  order  larger  than  that  one  chosen  for  the  original  system. 
The  reason  for  this  choice  will  be  explained  in  the  next  section.  In  the 
second  phase,  the  ME  of  the  original  system  are  considered.  Now,  the 
procedure  operates  iteratively:  in  the  first  step  the  higher  order  mom¬ 
ents  that  appear  in  the  equations  for  the  moments  of  k-th  order  are  con¬ 
sidered  as  known  quantities  taking  the  values  that  have  been  eval¬ 
uated  for  the  linearized  system.  In  the  generic  i-th  step  these  moments 
are  set  equal  to  the  values  obtained  in  the  step  z-1.  The  iterations  are 
carried  on  till  the  moments  do  not  change  appreciably.  This  procedure 
is  applied  to  a  Duffing  oscillator  excited  by  a  quadratic  polynonial  of  a 
filtered  Gaussian  process.  The  damping  and  the  nominal  vibration  pe¬ 
riod  of  the  oscillator  are  kept  constant,  while  the  bandwidth  of  the  fil¬ 
ter  is  varied  in  order  to  ascertain  the  effects  of  this  quantity  on  the  re¬ 
sponse.  The  results  of  the  proposed  approach  are  compared  with  those 
of  Monte  Carlo  simulation.  When  the  bandwidth  of  the  filter  is  narrow, 
eventual  multivalued  responses  are  searched  using  either  the 
stochastic  differential  calculus  or  Monte  Carlo  simulation. 


2.  PRELIMINARY  CONCEPTS 

In  the  study  of  nonlinear  vibrations  the  Duffing  equation  plays  a 
fundamental  role.  In  normalized  form  it  reads  as 

x(t)  +  f>Qx(t)  +  (£>Q^x(t)  +  sx(t)^'^  =  F(t)  (1) 

in  which  is  the  coefficient  of  viscous  damping,  cOq  is  the  pulsation  of 
the  corresponding  linear  oscillator  (e  =  0),  and  the  term  £x(  t  f  repre¬ 
sents  the  nonlinear  restoring  force,  acting  as  a  hard  or  soft  spring  for 
positive  or  negative  values  of  8,  respectively.  In  random  vibration 
studies  F(t)  is  generally  assumed  to  be  a  Gaussian  process,  since  this 
type  of  stochastic  process  allows  the  use  of  analytical  methods.  When 
Fit)  is  a  Gaussian  white  noise,  the  reduced  Fokker-Planck-Kolmogorov 
equation  [10]  associated  with  (1)  admits  an  analytical  solution  [24]  as 
follows 


1567 


p(x,x)  =  Cexp 


(2) 


in  which  p(x,x)  is  the  steady-state  joint  probability  density  function 
(PDF)  of  x,x}  C  is  a  normalization  constant,  and  zv,,  is  the  strength  of 
white  noise,  which  is  expressed  as  F(t)  =  ^KWQW( t ),  being  W(t)  a  unit 
strength  Gaussian  white  noise. 

However,  not  all  excitations  are  so  broad-banded  that  they  can  be 
idealized  as  a  white  noise.  Very  narrow-banded  excitations  can  be 
considered  as  monochromatic,  say 

F(t)  =  A(t)cos(0  ft  +  B(t)sin(ii  ft  (3) 

in  which  A(t)  and  B(t)  are  two  slowly  varying  Gaussian  processes. 
Otherwise,  a  frequent  idealization  of  F( t)  is  thinking  it  as  the  output  of 
a  second  order  linear  filter,  say 

F(t)~\jf(t)  yf(t)  +  ^  fyf(t)  +  (iijyf(t)  =  ^KZUQW(t)  (4) 

where  W(t)  is  a  unit  strength  Gaussian  white  noise.  The  steady-state 
FPK  equation  associated  with  the  augmented  system  of  (1)  and  (4)  has 
not  an  analytical  solution,  since  this  system  is  not  amenable  to  the 
classes  of  detailed  balance  or  of  generalized  stationary  potential  [10]. 
However,  the  primary  excitation  in  (4)  is  a  Gaussian  white  noise,  so 
that  the  problem  can  be  suitably  framed  in  the  context  of  Ito's 
stochastic  differential  calculus  [5-9]. 

Likewise  to  the  deterministic  response  of  Duffing  oscillator  sub¬ 
jected  to  a  sinusoidal  excitation,  in  which  there  appear  phenomena 
such  as  multiple  valued,  subharmonic  and  superharmonic,  responses 
and  the  associated  jumps  in  response  levels,  for  sufficiently  narrow 
bandwidths  of  the  excitation  the  random  response  can  exhibit  multiple 
values  of  the  statistical  moments,  particularly  of  the  variance.  This  fact 
was  revealed  by  the  analyses  performed  by  some  researchers  that  used 
different  approaches.  Some  used  equivalent  linearization  technique 
[13,14,25-27].  Lennox  and  Kuak  [28]  faced  the  problem  through  a 
combination  of  deterministic  and  stochastic  averaging.  Other  authors 
preferred  the  methods  of  multiple  time  scaling,  quasi-harmonic 
analysis  and  harmonic  balance  [29-31].  More  recently,  Ito  stochastic 
calculus  with  a  non-Gaussian  closure  was  used  in  Ref.  16.  However, 
the  significance  of  the  multiple  solutions  generated  by  equivalent 
linearization  and  Ito's  calculus  is  still  an  open  question.  The 
discussions  about  this  subject  in  Refs.  15, 16  are  remarkable. 

This  paper  concerns  the  response  of  Duffing  oscillator  (1)  to  an  ex¬ 
citation  of  the  form 

F(t)  =  aQ+aiyf(t)  +  a2yf(tf  (5) 


1568 


being  yjit)  the  output  of  a  filter  such  as  (4).  It  is  recalled  that  (5)  can 
represent  wind  excitation  by  properly  choosing  the  coefficients  a^-a^.  In 
a  literature  search  made  by  the  writers  no  theoretical  and  applied  stud¬ 
ies  have  been  encountered  regarding  the  response  of  Duffing  oscillator 
to  excitations  of  the  form  (5)  or  of  the  more  general  form 

Fit)  =  flo  +  a^y  fit)  ^  a^y  fit)^  ^ . (6) 

Clearly,  the  problem  is  rendered  more  difficult  by  the  twofold  non¬ 
linearity  in  both  the  structure  and  the  excitation.  If  Ito's  calculus  is 
applied,  the  ME  form  an  infinite  hierarchy  with  many  higher  order 
terms  arising  from  the  term  in  (1)  and  the  term  ajyj  in  (5).  In  this 
way,  a  large  number  of  cumulant-moment  relationships  must  be  set 
equal  to  zero  to  close  the  hierarchy,  making  the  ME  highly  nonlinear. 
Furthermore,  these  last  are  more  than  two  hundred,  if  the  statistical 
moments  of  fourth  order  are  to  be  computed.  Even  if  the  steady-state 
solution  is  of  concern,  and,  hence,  the  ME  become  algebraic,  it  is 
cumbersome  to  find  a  solution  and  more  physically  realizable 
solutions  may  exist.  In  order  to  overcome  these  difficulties,  advantage 
is  taken  from  the  fact  that  the  statistical  moments  of  a  linear  system 
excited  by  a  polynomial  form  of  a  filtered  Gaussian  process  are 
calculated  exactly  [2,  19-21].  Therefore,  in  the  first  step  of  the 
procedure  that  is  here  proposed  the  system  (1)  is  linearized.  The 
statistical  moments  that  are  computed  in  this  way  constitute  a  first 
estimate  of  the  moments  of  the  actual  system  and  are  the  basis  of  the 
algorithm  with  which  the  hierarchy  of  ME  is  closed. 


3.  ITERATIVE  PROCEDURE 
3.1  Linearized  system 

In  the  first  phase  of  the  iterarive  procedure  Eq.  (1)  is  substituted  by 
xit)+pQxit)  +  (i)^xit)  =  Fit)  (7) 

in  which  Fit)  is  given  by  (5),  and  is  the  linearization  parameter. 
Making  the  mean  square  error  between  (7)  and  (1)  minimum  [22,  23]  it 
is  got 

cOg  =o)o +ecoo  •e[x^]/e[x^]  (8) 

in  which  £[•]  denotes  ensemble  average.  Since  the  linearization  is 
confined  to  the  structure,  and  the  excitation  holds  the  nonlinear  form 
(5),  xit)  is  not  a  Gaussian  process  so  that  the  relationship  E[x^]  -  3(£[x‘]- 
E[x]f  is  not  valid.  Being  E[x‘‘]  unknown  a  priori,  the  equivalent 
linearization  is  applied  iteratively  starting  from  a  tentative  value  of  the 


1569 


fourth  moment  of  x.  The  statistical  moments  are  computed  to  the 
fourth  order,  and  a  new  value  of  co^"  is  calculated  by  using  (8).  The 

iterations  are  terminated  when  at  step  n  |cOg  „  -  co^ <8^,  being  5e  of 

the  order  10*^  It  is  recalled  that  the  non-Gaussianness  of  the  response 
makes  it  very  difficult  to  derive  an  analytical  relationship  for  response 
variance  as  it  is  done  in  Refs.  25,26.  Thus,  the  search  of  eventual 
multiple  values  of  variance  is  made  numerically. 

In  order  to  compute  the  statistical  moments  of  the  response,  Ito's 
differential  rule  is  applied  [6-9].  The  state  variables  x^-x,  X2-x  ,x^~ 
y^,  and  x^-y^  are  introduced  and  Eqs.  (4,7)  with  the  excitation  (5)  are 
written  in  incremental  form,  say 

dxi  =X2'dt  (9a) 

dx2  = -^qX2  - dt -(£)^x-^  ■  dt  +  Uq  ■dt  +  a-^x^  •  dt  +  ^2^3  •  dt  (9b) 
dx^  -x^-dt  (9c) 

dx4  =  -P  fX^  •  dt  -  C0yX3  •  dt  +  ■  dB  (9d) 


in  which  dB  is  the  increment  of  a  unit  Wiener  process  (Brownian  mo¬ 
tion)  that  is  related  to  white  noise  by  the  formal  relationship  dB/dt  = 
W(t). 

The  stationary  first  order  moment  (statistical  average)  of  x  is 
E[i]  =  E[j:i  ]  =  {uq  +  ]}  (10) 

E[x]  depends  on  the  mean  square  value  E[x3j  =  £[y?j,  that  is  known, 
being  y^  a  Gaussian  process. 

The  equations  for  the  stationary  second  order  moments  read  as 
2E[xiX2]  =  0  (11a) 

-2p  oE[^2  ]~  e  E[^i^2  ]  +  2aQE[x2]+  2aiE[x2X^  ]  +  2fl2E[^2^3  ]  =  0 

(11c) 

In  Eqs.  (11)  there  are  moments  of  order  lower  and  higher  than  the 
second.  The  former,  E[xi]  and  £[^2]/  are  known.  Viceversa,  the 

moments  of  the  types  E[ziZj^]  (2  =  1,2;  k  =  3,4),  and  j  (2  =  1,2; 

r+s  =  2)  are  unknown.  However,  Eqs.  11  do  not  constitute  an  infinite 
hierarchy  even  if  there  are  third  order  moments.  The  technique  pro¬ 
posed  by  Muscolino  [20]  is  used  to  solve  the  problem.  It  requires  the 
solution  of  two  linear  and  uncoupled  sets  of  equations  that  are  written 


1570 


for  the  moments  E[ziZj^],  and  respectively.  Analogously,  the 

evaluation  of  the  moments  of  third  and  fourth  order  of  xi,X2  requires 
the  solution  of  five  and  six  linear  uncoupled  sets  of  equations, 
respectively.  Once  convergence  is  obtained  for  co  J ,  the  moments  of  the 
linearized  system  are  computed  to  the  sixth  order.  In  this  way,  a  first  e- 
stimate  of  the  response  moments  is  obtained. 

3.2  Nonlinear  system 

Now  we  return  to  the  actual  system  driven  by  Eq.  (1).  Eqs.  (1,4)  are 
written  in  incremental  form:  Eq.  (11b)  is  replaced  by 

clx2  =  -p  0X2  ■  dt  -  (£)lxi  •  dt  -  ECO  gxf  ■  dt  +  Uq  ■  dt  +  ■  X2  •  dt  +  a2  ■  xl  •  dt 

(12) 

The  ME  of  the  actual  system  are  written  applying  Ito's  differential  rule 
again.  They  constitute  an  infinite  hierarchy  really,  since  in  the 
equations  for  the  moments  of  k-th  order  there  are  moments  of  order 
k+2  because  of  the  term  in  (12).  In  this  paper  a  new  technique  is 

used  to  close  the  hierarchy  profiting  by  the  solution  that  has  been 
obtained  for  the  linearized  sys  tern.  This  technique  operates  iteratively. 

Let  us  examine  some  ME  of  the  actual  system.  The  statistical  aver¬ 
age  is  given  by 

E[j:i]  =  E[x]  =  cOo^{ao  +fl2E[x|]}-eE[xq  (13) 

E[x]  depends  on  the  third  order  moment  j,  which  in  the  first  step 

is  assumed  to  be  known  and  take  the  value  that  has  been  computed  for 
the  linearized  system.  In  the  other  iterations  it  takes  the  value  of  the 
previous  one. 

The  equation  for  a  moment  of  order  m  can  be  written  as 
ii-pq  =^PP-p^l,q+l-^?>0\^pq-<^(^l\^p+l,q-l  '  (i^(^0P-p,q+2  +Wp.q-l  + 

in  which  p  +  q  =  m  and  yipcj-  structure  is  exposed  to  a 

stationary  excitation  for  a  sufficiently  long  time,  it  reaches  the  sta- 
tionarity  and  the  l.h.s.  of  (14)  vanishes.  In  Eq.  (14)  there  are  different 
types  of  moments:  The  fifth  addendum  has  an  order  lower  than  m  and 
has  been  already  evaluated,  while  the  first  three  addenda  are  of  the  m- 
th  order,  and  as  many  equations  as  these  moments  can  be  written. 


1571 


However,  the  fourth  term  and  the  last  two  are  additional  unknowns. 
The  cross-moments  among  x^,  and  the  variables  of  the  filter  are  eval- 
luated  writing  the  corresponding  equations  down,  as  for  the  linearized 
system,  and  the  hierarchy  is  overcome.  The  fourth  addendum  is 

an  effective  hierarchical  term.  In  the  first  step  of  the  iterative  proce¬ 
dure  it  takes  the  value  that  has  been  computed  for  the  linearized 
system,  while  in  the  generic  i-th  iteration  it  takes  the  value  of  the 
iteration  z-1.  In  this  way,  the  equations  for  the  moments  of  different 
order  and  type  are  considered  separately  with  an  evident  computa¬ 
tional  advantage.  Furthermore,  the  ME  remain  linear,  while  in  the 
cumulant-neglect  closure  method  [11,  12]  the  cumulant-moment  rela¬ 
tionships  introduce  non-linearities  of  degree  as  higher  as  the  level  on 
which  the  closure  is  made.  In  the  case  under  examination  with  four 
state  variables,  computing  the  moments  to  the  fourth  order,  there 
would  arise  more  than  two  hundred  equations,  and  this  large  nonlin¬ 
ear  set  of  equations  would  be  hardly  solvable  with  a  personal  comput¬ 
er.  Viceversa,  using  the  proposed  approach  the  response  moments  are 
computed  by  solving  linear  systems  with  a  small  number  of  equations. 
Since  the  highest  order  moment  appearing  in  (14)  has  the  order  7n+2 
and  the  moments  of  the  actual  system  are  computed  to  the  fourth 
order,  the  moments  of  the  linearized  system  are  evaluated  to  the  sixth 
order. 


3.3  Search  for  multiple  solutions 

Since  the  writers  did  not  find  any  study  on  Duffing  oscillator  ex¬ 
cited  by  a  polynomial  form  of  filtered  Gaussian  process,  it  was  not 
possible  either  to  affirm  or  to  exclude  the  possibility  of  a  multivalued 
variance  in  the  presence  of  a  narrow-band  excitation.  Hence,  there  is 
need  of  ascertaining  whether  the  response  moments  under  a  given 
excitation  are  unique  or  not.  The  polynomial  form  (5)  of  the  excitation 
does  not  leave  room  to  a  simple  expression  for  the  variance  as  in  the 
case  of  a  filtered  Gaussian  excitation  [13,14,25-27].  Moreover,  the 
bandwidth  of  the  excitation  cannot  appear  explicitly  in  this  case.  Thus, 
an  iterative-numerical  procedure  is  used  for  searching  multivalued 
responses. 

The  search  is  made  with  reference  to  the  linearized  system,  since  it 
is  simpler  to  be  done.  From  Eqs.  (11)  we  find  for  the  mean  square 
values  of  x^  and  x,,  respectively 

]  =  CO  7^  |e[x|  ]  +  aoE[x-^]+a^E[xiX3  ]  +  a2E^x-ixl  ]}  (15a) 


1572 


%!]=  Po^{^^iE[^2^3]  +  «2%24]}  (15b) 


Eq.  (8)  giving  the  linearization  parameter  is  written  as 
co^  =cOo|l.  +  £fcE[^i  ]j-  The  procedure  must  accomodate  for  the  possi- 

<2^ 

bility  of  a  multivalued  variance.  After  obtaining  convergence  for  cOg 
and  computing  the  statistical  moments,  the  ratio  k  =  £[Xj'']/E[x/]^  is 
calculated  and  substituted  in  the  expression  of  co  g ,  which  at  its  turn  is 
inserted  in  Eq.  (15a).  In  Eqs.  (15)  there  are  the  moments  £[xj,  E[xx^l 
and  E[xpc^]  (z=l,2)  that  obviously  depend  on  cOgi  As  a  first  approx¬ 
imation  these  moments  are  assumed  to  take  the  values  obtained  at  the 
end  of  the  iterations  for  finding  (^.  In  this  way  (15a)  becomes  a  quad¬ 
ratic  equation  in  the  unknown  £[Xi“].  Obviously,  the  two  roots  must  be 
real,  but  from  a  theoretical  point  of  view  it  carmot  be  excluded  that 
both  result  positive.  If  this  were  the  case,  the  iterative  procedure  would 
be  reentered  and  split  into  two  different  searches.  The  two  sets  of  sta¬ 
tistical  moments  that  would  be  computed  would  become  two  different 
bases  for  the  iterative  search  operated  on  the  actual  system.  In  this  way 
the  iterative  computation  of  the  moments  of  the  actual  system  becomes 
twofold,  but  a  -priori  it  is  not  possible  to  affirm  whether  the  con¬ 
vergence  is  towards  two  different  states  or  towards  the  global  state 
corresponding  to  the  unique  solution  of  the  FPK  equation.  The  re¬ 
search  for  a  multiple  solution  is  made  by  simulation  too,  as  will  be 
explained  in  next  section. 


4.  APPLICATIONS 

The  procedures  outlined  in  previous  sections  have  been  applied  to 
a  Duffing  oscillator  characterized  by  the  following  values  of  the 
parameters  in  Eq.  (1):  po  =  0.6283,  which  corresponds  to  a  ratio  of 
critical  damping  of  a  5  %;  coo  =  2tc;  £  =  0.5.  The  coefficients  of  Eq.  (5)  are: 
flg  =  10,  aj  =  =  1.  The  filter  (4)  is  characterized  by  co^  =  271,  and  =  5. 

As  regards  p^^  it  takes  the  values  0.1257,  0.3770, 1.2567,  which  correspond 
to  relative  dampings  of  0.01,  0.03,  and  0.10,  respectively.  The  bandwidth 
parameters  associated  with  these  two  values  are  0.1131,  0.1949,  and 
0.3386,  respectively.  The  bandwidth  parameter  is  defined  as 

=  being  Xj  the  spectral  moment  [32].  In  the  case 

of  Eq.  (4)  the  formula  q  =  (1.16^°-^^  -0.21)^-^  gives  an  accurate  estima¬ 
tion  [33].  It  is  remarked  that  the  set  of  parameters  co^  =  27C,  p^  =  0.1257, 


1573 


Table  1;  results  for  Duffing  oscillator^  0.1257 


Elx] 

■SSH 

(1) 

0.258459 

0.105127 

0.050338 

0.028410 

(2) 

0.273143 

0.073133 

0.024813 

0.008744 

(3) 

0.281231 

0.080166 

0.023150 

0.006769 

Stochastic  differential  calculus:  (1)  linearized  system,  (2)  actual  system. 

(3)  Simulation 


Table  2;  results  for  Duffing  oscillator,.  P,  =  0.3770 


e[x^] 

E[x^] 

(1) 

0.078837 

0.028604 

0.011598 

(2) 

0.253232 

0.073950 

0.023958 

0.008564 

(3) 

0.257975 

0.067416 

0.017316 

0.004505 

(l)-(3)  as  in  Table  1. 


Table  3;  results  for  Duffing  oscillator^  P,=  1.2567 


E[x] 

E[x^-] 

HBUII 

(1) 

0.246996 

0,064276 

0.017517 

0.004976 

(2) 

0.248568 

0.064627 

0.017491 

0.004913 

(3) 

0.254055 

0.064595 

0.016426 

0.004180 

(l)-(3)  as  in  Table  1. 


and  Wg  =  5  would  cause  a  multivalued  response  with  a  large 
probability  if  the  filter  output  were  directly  applied  to  the  structure. 
The  steady  state  response  only  is  considered  computing  the  statistical 
moments  to  the  fourth  order. 

The  results  obtained  by  the  use  of  the  procedure  proposed  in  this 
paper  are  checked  against  numerical  simulation.  It  has  not  been 
possible  to  perform  a  comparison  with  the  standard  cumulant-neglect 
closure  method,  for  both  the  hardware  and  the  software  available  to 
the  writers  were  inadequate  for  solving  a  nonlinear  algebraic  system  of 
more  than  two  hxmdred  equations.  Two  types  of  simulations  have  been 
performed,  the  former  of  which  is  a  standard  Monte  Carlo  simula¬ 
tion.  Samples  with  10,000  histories  of  motion  starting  from  zero  val¬ 
ues  of  both  initial  displacement  and  velocity  are  constructed  and 
analyzed  using  ordinary  statistical  tools.  The  motion  lasts  for  30  s  and 
Eq.  (1)  is  solved  by  applying  a  fourth  order  Runge-Kutta  scheme. 
However,  the  classical  simulations  cannot  reveal  multivalued 
responses,  since  the  averages  are  performed  on  a  large  number  of 


1574 


Fig.  1  -  Power  spectral 
density  of  the  output  of 
filter  (4),  13^=  0.1257. 


10.00  —1  y(t) 


0.00  400.00  800.00  1200.00  1600.00 


0.00  400.00  800.00  1200.00  1600.00 

Fig.  2  -  Top:  typical  time  history  of  the  output  of  filter  (4).  Bottom:  re¬ 
sponse  displacement  x{t),  (3^=  0.1257. 

samples.  Roberts  [15]  suggested  another  type  of  simulation,  in  which  a 
limited  number  of  motion  histories  is  simulated,  but  each  history  lasts 
for  a  relative  long  time  and  is  analyzed  alone.  Recently,  Cusumano  and 
Kimble  [34]  proposed  the  so-called  stochastic  interrogation  experimen- 
method,  in  which,  differently  from  Robert's  method,  the  histories  of 
motion  have  random  initial  conditions.  Furthermore,  Cusumano  and 
Kimble  consider  large  samples  and  trace  the  Poincare  maps  to  reveal 
the  presence  of  basins  of  attraction.  Due  to  hardware  limitations,  these 
two  types  of  simulation  are  performed  with  samples  of  10  histories  of 


1575 


1.20 
0.80  -1 
0.40 


x(t) 


0.00 
-0.40  H 


4- 


0.00  200.00  400.00 

Fig.  3  -  Response  displacement  x(t),  Py  =  0.3770. 


I  t(s) 
600.00 


motion.  When  the  initial  conditions  are  random,  the  initial  displace- 
cement  and  velocity  are  independent  normal  variates  with  zero  mean 
and  unit  variance. 

Tables  1-3  show  the  principal  statistical  moments  obtained  by  using 
the  three  methods,  for  =  0.1257,  0.3770,  and  1.2567,  respectively.  The 
statistical  linearization  fails  to  yield  accurate  estimates  of  the  statistical 
moments  for  the  case  p^  =  0.1257.  In  the  other  two  cases  the  results  for 
the  first  two  moments  are  quite  acceptable,  while  the  estimates  of  both 
the  third  and  fourth  moment  are  too  high.  In  general,  the  proposed 
approach  for  closing  the  hierarchy  of  the  ME  constitutes  a  substantial 
improvement  over  the  linearization  even  if  the  third  and  fourth 
moment  of  the  intermediate  case  are  greatly  overestimated  (by  a  38.4  % 
for  Efx^]  and  by  a  90  %  for  E[x^]).  Clearly,  as  higher  the  order  of  a 
moment  is,  as  worse  the  approximation  of  the  stochastic  differential 
calculus.  Using  a  higher  order  of  closure  should  improve  the  accuracy, 
but  this  requires  the  use  of  more  powerful  computers. 

The  performance  of  the  proposed  approach  is  noticeable  in  the  first 
case,  say  P^  =  0,1257.  This  case  is  characterized  by  a  very  narrow- 
banded  output  of  the  filter  (4)  (Fig.  1),  even  if  the  a  simulated  history 
of  yjit)  does  not  resemble  a  sinusoid  (Fig.  2,  top).  Performing  the  search 
for  multiple  solutions,  Eq.  (15a)  has  two  real  roots,  but  one  of  them 
only  is  positive.  Nevertheless,  examining  a  single  motion  history 
lasting  for  1,500  s  (Fig,  2,  bottom)  there  appears  a  strange  phenomenon. 
After  exposing  the  system  to  the  excitation  for  almost  400  s,  one  can 
note  that  evenly  spaced  peaks  rise,  whose  periodicity  is  90-100  s  about. 
During  these  peak  phases  the  oscillations  occur  around  the  unchanged 
mean  value,  but  their  amplitude  increases  time  after  time  and  seems  to 
be  unbounded.  The  writers  think  that  it  is  improper  to  define  this 
phenomenon  as  a  multivalued  response;  rather,  this  behavior  might  be 
assimilated  to  a  kind  of  instability.  An  ever  increasing  amplitude 
means  an  increasing  mean  square  value  that,  perhaps,  might  be 
detected  by  making  a  non  stationary  analysis.  Thus,  the  statistical 
moments  that  have  been  calculated  with  both  stochastic  differential 


1576 


calculus  and  simulation,  whose  motion  histories  last  for  30  s,  are 
correct  for  the  stationary  segment  of  the  response,  only.  The  same 
trend  can  be  noticed  for  the  case  =  0.3770,  but  it  is  slightly  faded. 


5.  CONCLUSIONS 

A  method  is  presented  to  close  the  hierarchy  of  the  ME  of  nonlinear 
systems  excited  by  polynomial  forms  of  filtered  Gaussian  processes. 
This  method  operates  in  two  phases  in  the  former  of  which  resort  is 
made  to  statistical  linearization  that  yields  a  first  estimate  of  the 
moments  of  the  non-Gaussian  response.  In  the  second  phase  the  ME  of 
the  actual  system  are  considered.  They  constitute  an  infinite  hierarchy 
to  close  which,  instead  of  using  the  classical  cumulant-neglect  closure 
method  that  may  be  cumbersome  from  a  computational  point  of  view, 
an  iterative  procedure  is  used.  This  is  based  on  the  estimate  of  the 
response  moments  that  has  been  obtained  for  the  linearized  system. 

The  proposed  method  is  applied  to  a  Duffing  oscillator  excited  by  a 
quadratic  polynomial  of  a  filtered  Gaussian  process.  The  bandwidth 
parameter  of  the  filter  is  varied  from  a  very  narrow  one  to  a  medium 
one  to  ascertain  whether  the  response  may  be  multivalued  as  in  the 
case  of  a  simple  narrow-banded  filtered  excitation.  In  the  case  of  a 
narrow-banded  excitation  a  phenomenon  different  from  those  reported 
in  literature  occurs.  After  a  certain  time  system  response  seems  to  lose 
the  stationarity  and  periodically  shows  peaks  with  ever  increasing 
amplitude.  This  phenomenon  cannot  be  detected  either  by  the 
stochastic  differential  calculus,  if  the  ME  are  solved  for  the  steady 
state,  or  by  the  customary  simulation  in  which  the  histories  of  motion 
have  a  brief  duration.  In  order  to  reveal  this  behavior  it  is  necessary  to 
study  responses  lasting  some  hundreds  of  seconds.  However,  the 
response  moments  that  are  given  by  the  proposed  approach  are 
substantially  correct  for  the  stationary  segment  of  the  response. 

ACKNOWLEDGEMENTS 

The  authors  are  indebted  to  Prof.  Mario  Di  Paola  of  the  University  of 
Palermo,  Italy,  for  his  continuous  advice  and  encouragement. 

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1579 


1580 


EXTREME  RESPONSE  ANALYSIS  OF  NON-LINEAR 
SYSTEMS  TO  RANDOM  VIBRATION 


S.  McWilliam 

Department  of  Mechanical  Engineering,  University  of  Nottingham, 
Nottingham,  NG7  2RD,  England. 

ABSTRACT 

This  paper  considers  the  numerical  solution  of  the  Fokker-Planck-Kolmogorov 
(FPK)  equation  which  governs  the  transition  probability  density  function  (pdf) 
for  a  certain  class  of  non-linear  system.  In  order  to  provide  a  numerical  solution 
for  the  pdf  which  is  well  behaved  at  its  tails  (and  thus  better  suited  to  extreme 
value  problems),  the  FPK  equation  is  transformed  to  give  an  equation  for  the  log- 
pdf.  The  resulting  non-linear  partial  differential  equation  is  solved  using  a 
weighted-residual  approach.  This  technique  is  applied  to  first  and  second  order 
systems,  including  the  Duffing  oscillator.  The  accuracy  of  the  technique  is 
assessed  by  comparison  with  analytic  solutions  and  numerical  simulation. 

1.  INTRODUCTION 

Many  different  types  of  structure,  such  as  offshore  structures,  are  subjected  to 
random  loading.  In  the  design  of  these  structures  it  is  particularly  important  that 
the  statistics  of  the  response  are  obtained  so  that  the  maximum  stresses  and 
fatigue  life  of  the  structure  can  be  predicted.  For  the  situation  when  the  system 
is  linear  and  the  excitation  is  Gaussian  the  response  statistics  are  well-known. 
However,  most  practical  engineering  structures  are  non-linear  to  some  extent  and 
the  problem  of  predicting  the  response  statistics  is  much  more  difficult. 

For  systems  which  are  subjected  to  white-noise  excitation,  the  response  is  a 
Markov  process  and  the  transitional  probability  density  function  (pdf)  is  governed 
by  the  Fokker-Planck-Kolmogorov  (FPK)  equation.  For  other  systems  the  FPK 
equation  can  be  applied  provided  that  the  response  is  a  higher  order  Markov 
process.  In  both  these  cases  the  problem  of  predicting  the  response  statistics 
reduces  to  solving  the  FPK  equation  for  the  transitional  probability  density 
function.  The  difficulties  associated  with  finding  exact  solutions  to  this  equation 
have  lead  a  number  of  authors  to  develop  approximate  solution  procedures  [1-3]. 
Although  these  techniques  are  limited  to  the  analysis  of  low  order  systems,  they 
have  been  applied  to  a  number  of  practical  engineering  problems  (e.g.[3,4]). 

The  most  commonly  used  numerical  solution  procedures  are  the  Galerkin 
technique  [1],  the  Finite  Element  method  [2]  and  the  path-integral  method  [3]. 


1581 


Wen  [1]  developed  a  Galerkin  method  for  the  solution  of  the  non-stationary  FPK 
equation,  in  which  the  joint  probability  density  function  was  expanded  in  terms 
of  Hermite  polynomials.  The  disadvantages  of  this  technique  are  that  it  tends  to 
have  a  low-rate  of  convergence  for  systems  which  are  strongly  non-linear  and  the 
accuracy  of  the  tails  of  the  distribution  is  not  guaranteed  -  yielding  negative 
values  .  In  contrast,  the  Finite  Element  method  [2]  provides  a  more  robust 
numerical  solution  procedure  which,  in  principle,  has  the  potential  to  deal  with 
all  types  of  non-linearity  and  give  an  accurate  prediction  of  the  tails  of  the 
distribution.  However,  a  large  number  of  elements,  and  hence  a  large  amount  of 
cpu  time,  are  needed  to  give  sufficient  resolution  over  the  tails  of  the  distribution 
in  order  for  it  to  be  used  for  the  prediction  of  extremes.  The  path-integral  method 
proposed  by  Naess  [3]  has  recently  been  shown  to  provide  an  accurate  means  of 
estimating  the  response  pdf,  including  the  tails.  However,  this  too  requires  a 
substantial  amount  of  cpu  time,  and  is  very  much  at  a  developmental,  stage. 

In  contrast  to  the  above  techniques,  the  primary  aim  of  the  present  work  is  to 
accurately  estimate  the  tails  of  the  response  distribution.  This  is  achieved  by 
considering  the  log-pdf,  which  has  the  advantage  of  ensuring  that  the  resulting 
probability  density  function  is  always  positive.  This  approach  has  recently  been 
applied  by  Di  Paola  et  al.  [5]  in  which  the  log-pdf  was  expanded  as  a  Taylor 
series.  For  the  examples  considered  there,  which  included  various  first  order 
systems  and  the  Duffing  oscillator,  it  was  shown  that  this  technique  gave  good 
agreement  over  the  main  body  of  the  pdf,  but  no  results  for  the  tails  of  the 
distribution  were  reported.  A  disadvantage  of  the  adopted  Taylor  series  solution 
method  is  that  it  can  not  be  used  to  analyse  systems  which  contain  non-linearities 
whose  derivatives  are  discontinuous  at  the  origin  (e.g.  x\x\). 

In  this  paper  an  approach  similar  to  that  proposed  by  Di  Paola  is  used  in  which 
an  orthogonal  expansion  for  the  log-pdf  is  obtained.  More  specifically,  a  Hermite 
polynomial  expansion  of  the  log-pdf  is  obtained  by  using  a  weighted-residual 
approach  to  solve  the  governing  differential  equation  for  the  log-pdf.  The  main 
advantage  of  using  this  approach  compared  to  a  Taylor  series  solution  is  that  it 
is  reasonably  easy  to  incorporate  non-linearities  such  as  x|x|  into  the  analysis. 
The  accuracy  of  the  proposed  numerical  solution  procedure  to  predict  the 
response  statistics  of  various  first  and  second  order  systems  is  investigated  by 
comparison  with  analytic  results  or  numerical  simulation. 

2.  THE  FPK  EQUATION  AND  LOG-PDF  EQUATION 

Many  references  (for  example  [6])  give  details  of  the  derivation  of  the  Fokker- 
Planck-Kolmogorov  (FPK)  equation  for  the  determination  of  the  transient 
probability  density  function  (pdf)  of  non-linear  systems  subjected  to  random 
excitation.  Here,  the  FPK  is  simply  stated  and  then  used  to  derive  the  associated 


1582 


log-pdf  equation. 

In  order  to  apply  the  FPK  equation  it  is  necessary  to  write  the  equations  of 
motion  of  the  system  in  state-space  notation  as  follows: 

z  =  g(z)-^Aw,  (1) 


where  z  is  a  vector  containing  the  displacements  and  velocities  of  the  system,  A 
is  a  square  matrix  (assumed  here  to  be  constant),  g(z)  is  a  general  vector  function 
of  the  variables  z,  and  is  a  vector  of  uncorrelated  Gaussian  white  noise 
processes,  each  having  a  spectral  density  of  unity. 


If  z  constitutes  a  Markov  process,  the  FPK  equation  which  governs  the 
transitional  probability  density  function  p(z,t)  is  given  by  [6]; 


dp(z,0  _ 
dt 


'  dz.dz 


(2) 


where  n  is  the  dimension  of  z  and  JS  is  a  matrix  given  by: 

B=2tiAA  (2) 

As  mentioned  in  the  introduction,  equation  (2)  can  be  solved  numerically  using 
the  Galerkin  method,  the  Finite  Element  method  and  the  path-integral  technique. 

In  order  to  obtain  an  equation  which  is  more  suitable  for  determining  the  tails  of 
the  response  distribution  (and  hence  extremes)  it  is  convenient  here  to  express  the 
probability  density  function  (pdf)  as  follows: 

p(z,0=exp(p(z,0),  (4) 

where  p  is  the  log-pdf,  i.e.: 

p(z,0=ln;?(z,0- 


Differentiating  equation  (5)  with  respect  to  t  gives: 

ap(z,0  _  1  dpjzd)  (^5^ 

dt  piz,t)  dr 

Using  equations  (4)  and  (6),  equation  (2)  can  be  re-written  in  terms  of  the  log-pdf 
p(z,t)  as  follows: 


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dt  ,=i 


dz.  ^‘dz. 


I  1  ^  W 
/  /  =  1  ;=1 


0p  6p 


dz  .dz .  dz  .dz. 


(7) 


Equation  (7)  is  a  non-linear  partial  differential  equation  governing  the  transient 
log-pdf,  and  forms  the  basis  of  the  technique  proposed  here.  In  what  follows  a 
weighted  residual  procedure  for  solving  this  equation  is  presented. 

3  WEIGHTED  RESIDUAL  SOLUTION  FOR  THE  LOG-PDF 

The  weighted  residual  solution  of  equation  (7)  is  considered  for  a  first  order 
system  and  then  for  a  second  order  system. 

3.1  First  Order  Systems 

For  the  case  of  first  order  systems  equation  (7)  can  be  written  as  follows: 


0^p(z,r) 

f  ap(z,oy 

dt  dz  dz 

dz'^ 

[  dz  J 

In  the  weighted  residual  approach  adopted  here  the  log-pdf  and  the  system  non¬ 
linearity  are  expressed  as  the  sum  of  a  finite  number  of  weighted  Hermite 
polynomials,  such  that: 


m 


p(z,f)=EA.,(/)H,(z), 

/=0 


(9,10) 


where 


f g(z)ff^(z)e 


(11) 


and  Hi{z)  is  the  f  th  Hermite  polynomial. 

Substituting  equations  (9)  and  (10)  into  equation  (8)  gives: 


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(12) 

Multiplying  equation  (12)  by  /f*exp(-zV2)// 27i:,  integrating  over  z  from  -«>  to  ~ 
and  using  the  properties  of  the  Hermite  polynomials  given  in  Appendix  A  gives: 


k\\  =  ~k\ i “  1  ^ 


1=0  /=o 


yt=0, 


(13) 


1=0  y=o 


where  /[zj,  A;]  is  an  integral  of  a  triple  Hermite  product  which,  using  the  properties 
given  in  Appendix  A,  may  be  shown  to  be  given  by: 


npM-jlT. 


{ 

i 

! 

'  UL-k 

I  2  2  2j 

[  2  2  2) 

where  k\s  the  smallest  of  ij  and  k. 


(14) 


Equation  (13)  represents  a  set  of  first  order,  non-linear,  differential  equations 
which  can  be  solved  numerically  for  X*(0  using  standard  integration  techniques. 
The  calculated  values  can  then  be  substituted  back  into  equation  (9)  to  yield  an 
expression  for  p{z).  To  ensure  that  this  is  a  valid  pdf,  the  coefficient  is 
modified  to  ensure  that  the  normalisation  condition  is  satisfied. 


For  the  situation  when  the  response  of  the  system  is  stationary  (ie.  dA,k/dt=0  for 
all  k),  equation  (13)  reduces  to  a  set  of  non-linear  algebraic  equations.  Although 
the  solution  of  these  equations  can,  in  principle,  be  calculated  using  standard 
numerical  methods,  these  methods  require  a  good  first  estimate  of  the  solution  in 
order  to  achieve  a  good  rate  of  convergence.  For  the  systems  considered  in  the 
numerical  examples  section,  the  stationary  pdf  is  calculated  by  integrating 
equation  (12)  until  stationary  conditions  are  achieved. 

This  section  has  shown  how  a  weighted  residual  approach  can  be  used  to  obtain 
a  numerical  solution  to  the  log-pdf  equation  for  first  order  systems.  In  what 


1585 


follows  this  technique  is  extended  to  deal  with  second  order  systems. 

3.2  Second  order  systems 

An  identical  procedure  to  that  presented  for  first  order  systems  is  now  applied  to 
a  general  second  order  system. 

Let  the  log-pdf  be  expressed  as  follows: 

n  m 

p(z,.z,,/)  (15) 

p=l  q  =  i 

and  the  system  non-linearity  be  expressed  as: 

E  (>6) 

1=0  «=0 


where 


1 


27r/!m! 


and  Hj  is  the  z’th  Hermite  polynomial. 

A  set  of  non-linear,  first  order  differential  equations  in  terms  of  can  be 
obtained  by  substituting  equations  (15)  and  (16)  into  equation  (7),  multiplying  by 
Hp(z  i)H^{z2)Qxp{~{z^^  +Z2^)/2)/(27c),  integrating  over  2,  and  from  -<»  to  and 
making  use  of  equation  (14).  This  yields  an  equation  of  the  form: 

^M^E^A+EE^taV/-’  (18) 

ij  ij  l.m 

where  a,y  and  6  j,/„,are  constant  coefficients  dependent  upon  the  system  considered. 
As  in  section  3.1,  equation  (18)  can  be  solved  numerically  for  Xp^  and  then  the 
log-pdf  can  be  obtained  from  equation  (15).  This  procedure  is  used  in  the 
following  section  to  calculate  the  response  statistics  of  some  example  systems. 
In  addition,  it  may  be  noted  that  the  extension  of  the  weighted  residual  technique 
to  multi-dimensional  systems  is  straight-forward. 


1586 


4  NUMERICAL  EXAMPLES 


The  techniques  described  above  are  used  here  to  determine  the  stationary 
response  statistics  of:  i)  first  order  systems  and  ii)  second  order  systems.  In  all 
cases  the  initial  response  conditions  are  assumed  to  have  a  Normal  distribution 
and  the  stationary  response  statistics  are  obtained  by  integrating  the  equations 
governing  the  Hermite  coefficients  until  stationary  conditions  are  obtained.  The 
response  pdf  can  then  be  obtained  by  substituting  the  Hermite  expansion  for  the 
log-pdf  into  equation  (4)  and  applying  the  normalisation  condition  (to  ensure  that 
it  is  a  valid  probability  density  function). 

4.1  First  Order  Systems 

Two  first  order  systems  are  considered  here.  The  first  consists  of  a  system  with 
a  polynomial  non-linearity,  while  the  second  consists  of  a  drag  non-linearity.  In 
both  cases  the  results  obtained  using  the  proposed  technique  are  compared  with 
the  exact  analytic  solutions  available,  and  both  the  pdf  and  log-pdf  of  the 
response  (z)  are  plotted. 

4.1.1  Polynomial  non-linearity 

The  first  order  system  considered  here  has  the  following  equation  of  motion: 

x  =  -'^ax‘+M  (19) 


where  a,=0.1,  ^2=0.0,  ^3=0.05,  af=0.01,  and/(0  is  a  zero  mean,  Gaussian, 

white-noise  excitation  with  a  constant  spectral  value  K=1/'k.  The  stationary 
distribution  in  this  case  is  given  by: 

5 

p(x)  =Aqxp(-2  '^^)  (2^) 


where  A  is  the  normalisation  coefficient. 

The  distribution  of  x  for  this  case  is  shown  in  Figure  1,  where  Figure  la)  shows 
the  pdf  and  Figure  lb)  shows  the  log-pdf  In  these  figures  a  comparison  with  the 
exact  result,  given  by  equation  (20),  is  made  with  the  results  obtained  using  the 
proposed  technique  with  m=4  and  m=6  in  equation  (9),  and  an  equivalent 
Gaussian  distribution.  In  both  cases  it  is  seen  that  the  response  statistics  are 
highly  non-Gaussian,  and  the  proposed  technique  gives  better  agreement  with  the 
exact  result  with  m=6  than  with  m=4.  Further,  for  the  case  when  mi 6  it  is  found 
that  the  proposed  technique  agrees  exactly  with  the  exact  result.  In  many  respects 


1587 


this  agreement  obtained  using  the  proposed  technique  is  not  surprising  since  the 
stationary  pdf  can  be  expressed  exactly  in  the  assumed  form  when  ms  6. 
However,  there  is  no  guarantee  that  the  transient  response  can  be  expressed  in  this 
assumed  form. 

4. 1 .2  Drag  non-linearity 

The  system  considered  here  has  the  following  equation  of  motion: 

x--x\x\+f{t),  (21) 

where  f{t)  is  a  zero  mean,  Gaussian,  white-noise  excitation  with  a  constant 
spectral  value  K=\!tz.  The  exact  distribution  in  this  case  is  given  by: 

p(x)=^exp(^-^),  ‘  (22) 


where  A  is  the  normalisation  coefficient. 

Unlike  the  system  considered  in  section  4.1.1,  the  drag  non-linearity  and  log-pdf 
can  not  be  expressed  exactly  using  Hermite  polynomials,  and  is  therefore  a  more 
difficult  system  to  analyse  using  the  proposed  technique. 

Figure  2  shows  the  statistical  distribution  of  x  for  this  situation,  where  Figure  2a) 
shows  the  pdf  and  Figure  2b)  shows  the  log-pdf  In  these  figures  a  comparison 
with  the  exact  result,  given  by  equation  (22),  is  made  with  the  results  obtained 
using  the  proposed  technique  with  m=3  and  m=7  in  equation  (9),  and  an 
equivalent  Gaussian  distribution.  It  is  seen  that  the  response  statistics  are  highly 
non-Gaussian  and  that  using  an  increased  number  of  Hermite  polynomials  to 
represent  the  log-pdf  gives  improved  agreement  with  the  exact  result,  especially 
at  the  “tails”  of  the  distribution.  This  is  not  surprising  since  the  technique 
proposed  here  deals  with  the  log-pdf  and  is  therefore  well-suited  to  determining 
the  tails  of  the  distribution. 

4.2  Second  Order  Systems 

From  a  practical  point  of  view  it  is  the  response  statistics  of  second  (and  higher) 
order  systems  which  are  of  vital  importance  to  design  engineers.  In  order  to 
demonstrate  the  performance  of  the  proposed  technique,  two  second  order 
systems  are  considered.  In  each  case  results  are  obtained  for  the  displacement  (^i) 
pdf  and  the  velocity  (zj)  pdf,  and  the  results  are  plotted  as  a  pdfs  and  log-pdfs. 

4.2.1  Duffing  Oscillator 

The  equation  of  motion  considered  in  this  case  is  given  by: 


1588 


(23) 


where  /  (t)  is  a  zero  mean,  Gaussian,  white-noise  excitation  with  a  constant 
spectral  value  K=1/ti.  In  this  case  it  is  well-known  [6]  that  the  displacement  and 
velocity  are  statistically  independent,  and  exact  analytic  results  are  available  for 
the  displacement  and  velocity  pdf  s,  such  that: 

p(z,)=C,exp(-^z,^-iz,‘'),  p(z2)=C,exp(-izj\  (24,25) 


where  Ci  and  C2  are  normalisation  constants. 

The  stationary  response  statistics  for  this  case  are  shown  in  Figure  3,  where  the 
analytic  results  have  been  compared  with  those  obtained  using  the  proposed 
technique  and  an  equivalent  Gaussian  distribution.  In  this  case  it  is  found  that  the 
proposed  technique  gives  identical  results  for  the  jpdf  (and  hence  displacement 
and  velocity  statistics)  provided  that  a  sufficient  number  of  Hermite  polynomials 
are  used  (i.e.  n^4  and  m^S  in  equation  (15)).  In  some  respects  this  is  not 
surprising  since  it  is  known  that  the  stationary  jpdf  can  be  expressed  exactly  in 
the  assumed  form  given  by  equation  (15),  provided  a  sufficient  number  of  terms 
are  used.  However,  as  in  previous  examples,  it  is  unlikely  that  the  transient 
response  statistics  can  be  expressed  exactly  in  this  form. 

4.2.2  Duffing  Oscillator  with  cubic  damping  non-linearity 
The  equation  of  motion  considered  in  this  case  is  given  by: 

z^  =  -z2,  =  (26) 


where  f  {t)  is  a  zero  mean,  Gaussian,  white-noise  excitation  with  a  constant 
spectral  value  ^^=1/71:.  Although  there  is  no  exact  analytic  result  available  for  the 
displacement  distribution  in  this  case,  it  is  shown  in  Appendix  B  that  there  exists 
an  analytic  result  for  the  velocity  statistics,  such  that: 


piz^)=Atxp(-—-j^), 


(27) 


where  A  is  the  normalisation  constant. 

The  stationary  response  statistics  for  this  case  are  shown  in  Figure  4,  where  a 


1589 


comparison  is  made  with  numerical  simulation  and  exact  results.  In  the  proposed 
representation  n=4  and  m=8  are  used  in  equation  (15),  and  an  equivalent 
Gaussian  distribution  is  shown  for  comparison.  Up  to  the  highest  levels  obtained 
by  simulation,  the  proposed  method  gives  excellent  agreement. 

5  CONCLUSIONS 

A  numerical  method  for  solving  the  associated  FPK  equation  governing  the 
response  statistics  of  non-linear  systems  subjected  to  random  vibration  has  been 
presented.  The  main  advantage  of  this  method  over  alternative  techniques  is  that 
it  ensures  that  the  tails  of  the  response  distribution  remain  positive,  suggesting 
that  it  well-suited  to  extreme  value  prediction.  For  the  numerical  examples 
considered  this  was  shown  to  be  the  case,  in  which  excellent  agreement  with 
exact  analytic  solutions  and  numerical  simulation  were  obtained,  especially  at  the 
tails  of  the  distribution.  Although,  in  principle,  the  procedure  developed  is 
applicable  to  an  ^2-dimensional  system,  the  computer  costs  increase  rapidly  with 
n.  However,  the  remarkable  increases  in  computer  performance  over  the  past 
decade  suggests  that  the  present  procedure  may,  in  the  near-future,  become  a 
viable  option  for  calculating  the  response  statistics  of  higher  order  systems. 

REFERENCES 

1.  Wen,  Y.K.  Approximate  method  for  nonlinear  random  vibration. 
Proceedings  of  the  American  Society  of  Civil  Engineers,  Journal  of  the 
Engineering  Mechanics  Division,  1975, 101,389-401. 

2.  Langley,  R.S.  A  finite  element  method  for  the  statistics  of  non-linear 
random  vibration.  Journal  of  Sound  and  Vibration,  1985, 101(1),  41-54. 

3.  Naess,  A.  and  Johnsen,  J.M.  Response  statistics  of  nonlinear,  compliant 
offshore  structures  by  the  path  integral  solution  method.  Probabilistic 
Engineering  Mechanics,  1993,  8,  91-106. 

4.  Roberts,  J.B.  A  stochastic  theory  for  nonlinear  ship  rolling  in  irregular 
seas,  Journal  of  Ship  Research,  1982, 26,  229-245. 

5.  Di  Paola,  M.,  Ricciardi,  G.,  and  Vasta,  M.  A  method  for  the  probabilistic 
analysis  of  nonlinear  systems,  Probabilistic  Engineering  Mechanics, 
1995, 10(1),  1-10. 

6.  Lin,  Y.K.  Probabilistic  Theory  of  Structural  Dynamics,  McGraw-Hill, 
New  York,  1967. 


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APPENDIX  A  -  PROPERTIES  OF  HERMITE  POLYNOMIALS 
The  following  properties  of  the  set  of  Hermite  polynomials  are  used  in  section  3. 


^  dz 


=0  «=0 

-n\ 


(Al) 


dj^ 

dz 


=nH^_^iz). 


(A2) 


APPENDIX  B  -  ANALYTIC  SOLUTION  FOR  THE  VELOCITY 
STATISTICS  OF  A  PARTICULAR  CLASS  OF  SECOND  ORDER 

SYSTEM. 


For  the  second  order  systems  considered  in  the  numerical  examples  section  the 
associated  FPK  equation  (equation  (2))  can  be  written  as  follows: 


dp{z^,z^)  _  Bp{z^,z^  dg{z^,z^p{z^,z^ 

dt  ^  5z,  dz^  dz'} 


(Bl) 


where 


(B2) 


Substituting  equation  (B2)  into  equation  (Bl)  and  integrating  over  Zj  from  -oo  to 
gives: 


dt  ~ 


dz^ 


/ 


5p(2,,0  , 


az^^ 


(B3) 

The  first  term  on  the  right  hand  side  of  equation  (B3)  will  reduce  to  zero  since  for 


1591 


it  is  reasonable  to  assume  that  as  the  displacement  (z,)  increases  (or  decreases) 
without  bound p{z^,z^  will  tend  to  zero. 


Provided  that  is  continuous,  the  third  term  on  the  right  hand  side  of 

equation  (B3)  can  be  re-written  as: 


■■  — — 


(B4) 


The  term  in  square  brackets  on  the  right  hand  side  of  this  equation  will  reduce  to 
zero  provided  that  g,  is  an  odd  function  and  p{z^,Zj)  is  symmetric  with  respect  to 
Zi  for  all  values  of  This  condition  is  satisfied  by  the  systems  considered  in 
sections  4.2.1, 4.2.2  when  the  response  is  stationary  (i.e.  6p/9t=0).  Consequently, 
for  the  examples  considered  in  the  numerical  examples  section  the  third  term 
appearing  in  equation  (B3)  reduces  to  zero  provided  that  the  term  on  the  left 
hand-side  is  set  to  zero  (i.e.  stationary  conditions). 

Making  use  of  the  above  observations  and  re-arranging,  equation  (B3)  can  be 
written  as: 

(B5) 

5^2  dzl 


This  is  the  associated  stationary  FPK  equation  for  the  first  order  system  whose 
equation  of  motion  is: 

(B6) 


Thus,  it  has  been  shown  that  the  stationary  distribution  Zi  (alone)  is  identical  to 
that  of  a  one-dimensional  Markov  process.  Further,  the  solution  of  equation  (B5) 
can  be  written  as; 


2  f 

p{zf,  =A  exp(  j  gfz)dz), 


(B7) 


where  A  is  the  normalisation  constant.  Consequently,  the  velocity  statistics  of  the 
second  order  systems  considered  in  the  numerical  examples  section  can  be 
obtained  analytically.  This  result  is  used  to  validate  the  accuracy  of  the  results 
in  the  numerial  examples  section. 


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Normalised  displacement  pdf 


Figure  1 .  The  stationary  response  statistics  of  the  system  given  by  equation  (19). 

— ,  exact  solution;  -  •  -  ,  proposed  technique  (m=4); . ,  equivalent  Gaussian 

distribution,  (a)  displacement  pdf,  (b)  displacement  log-pdf  (Note:  the  exact 
solution  and  proposed  solution  (m=6)  agree  exactly  in  this  case^ 


Figure  2.  The  stationary  response  statistics  for  the  system  given  by  equation  (21): 
— ,  exact  solution;  -  •  -  ,  proposed  technique  (m=3);  •  •  proposed  technique 
(m=7);  •••••  ,  equivalent  Gaussian  distribution,  (a)  displacement  pdf,  (b) 
displacement  log-pdf 


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Normalised  velocity  pdf  Normalised  displacement  pdf 


Figure  3a) 


Figure  3b) 


Normalised  displacement  Normalised  displacement 


Figure  3c)  Figure  3d) 


Figure  3.  The  stationary  response  statistics  for  the  Duffing  Oscillator  given  by 

equation  (23); - ,  exact  solution;  -  •  - ,  proposed  technique; . ,  equivalent 

Gaussian  distribution,  (a)  displacement  pdf,  (b)  displacement  log-pdf,  (c)  velocity 
pdf,  (d)  velocity  log-pdf.  (Note:  the  exact  solution  and  proposed  solution  agree 
exactly  in  this  case.) 


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Normalised  displacement  pdf 


Figure  4c) 


Figure  4d) 


Figure  4.  The  stationary  response  statistics  for  the  Duffing  Oscillator  with  cubic 

damping  (see  equation  (26)); - ,  numerical  simulation  ((a)  and  (b)  only);  -  • 

proposed  technique;  -  -  ,  exact  solution  ((c)  and  (d)  only);  •••••  ,  equivalent 
Gaussian  distribution,  (a)  displacement  pdf,  (b)  displacement  log-pdf,  (c)  velocity 
pdf,  (d)  velocity  log-pdf. 


1595 


On  the  use  of  Finite  Element  solutions  of  the  FPK  equation 
for  non-linear  stochastic  oscillator  response  statistics 

M.  Ghanbari*  and  J.  F.  Dunne 

School  of  Engineering,  University  of  Sussex, 

Palmer,  Brighton,  U.K.  BNl  9QT. 

SUMMARY 

Numerical  solutions  of  the  stationary  Fokker-Planck  equation  are  obtained  via 
Langley's  FE  method  and  applied  to  non-linear  oscillators  driven  externally  by 
Gaussian  white-noise.  A  SDOF  model,  appropriate  to  large  amplitude  random 
vibration  of  a  clamped-clamped  beam  excited  by  band-limited  noise,  is  used  to 
demonstrate  convergence  of  predicted  response  marginal  density  functions  and 
extreme-value  exceedance  probabilities.  Experimental  verification  ol  this  beam 
model  is  shown  at  marginal  density  level  using  calibrated  FEM-FPK 
predictions  and  measured  data.  Convergence  of  a  4D  FEM-FPK  version  is  also 
demonstrated  at  marginal  density  level,  corresponding  to  a  pair  of  non-linear 
osciUators.  This  4D  method  is  appUed  up  to  a  practical  storage  limitation 
imposed  using  a  systematic  FEM  node  numbering  scheme. 


LO  INTRODUCTION 

Finite  element  analysis  has  now  found  widespread  application  in  the  field  of 
stochastic  dynamics  for  mathematical  modelling  of  structures  exposed  to 
random  type  loading.  Applications  vary  from  wave  induced  response 
prediction  for  fixed  and  compliant  offshore  structures,  seismic  response 
modelling  of  building  structures  during  earthquakes,  wind  buffeting  problems  in 
aerodynamics,  and  of  course  for  modeUing  the  effects  of  roughness  on  road  and 
rail  vehicular  motions  [1][2]. 

Predictions  of  reliability,  in  the  form  of  appropriate  response 
statistics,  can  usually  be  obtained  for  linear  structures  from  a  finite  element 
structural  model,  combined  with  linear  statistical  and  normal  distribution  theory 
[3].  In  cases  where  significant  nonlinearities  are  present,  accurate  response 
statistics  cannot  be  obtained  with  linear  theory,  justifying  alternatives  such  as 
Monte  Carlo  simulation  in  the  time-domain.  Simulations  can  be  very  effective 
in  obtaining  low-level  statistics,  for  example  the  first  few  response  moments 
(even  when  there  are  many  degrees-of-freedom).  But  to  obtain  highly  accurate 
information  in  the  tails  of  the  marginal  response  amplitude  probability  density 
iunction  (pdf),  or  to  obtain  accurate  low-level  extreme-value  exceedance 
probabilities,  conventional  simulations  are  impractical  owing  to  the  enormously 
Ions  runs  needed  to  obtain  confident  probability  estimates.  Response  amplitude 
proWbilities  are  useful  for  initial  reliability  assessment,  whereas  extreme- value 


■  Now  at:  RHP  BEARINGS  Ltd,  01dend.s  Lan^,|^(^ehouse,  Gloucestershire.  GLIO  3RH  UK. 


statistics  are  very  important  for  quantifying  the  likelihood  of  dangerously  high 
amplitudes  being  reached  within  a  specified  period  of  time,  since  these  can 
often  be  related  directly  to  risks  elsewhere. 

Focusing  initially  on  extreme-value  statistics,  a  number  of 
vibration  type  approaches  have  been  developed  using  a  combination  of 
vibration  theory,  control  techniques  and  simulation  [4-8].  These  methods  are, 
under  certain  conditions,  capable  of  producing  highly  accurate  results,  although 
for  broad-band  excitation,  there  is  still  a  need  for  a  direct,  accurate,  and 
efficient  approach  which  does  not  involve  complicated  intermediate  stages  or 
simulation.  More  generally,  within  the  theory  of  stochastic  processes,  there  are 
several  important  asymptotic  routes  to  extreme- value  prediction  appropriate  for 
high  response  amplitude  levels  or  long  sample  path  durations.  Local  maxima 
statistics  for  example,  can  be  used  to  construct  extreme-value  statistics, 
assuming  independence  of  specified  response  maxima  [9]  -  an .  approach 
frequently  used  for  linear  structures  with  Gaussian  responses.  Another 
approach  [8]  uses  the  asymptotically  Poisson  character  of  local  maxima  at  high 
levels.  An  alternative  route,  via  threshold  crossing  statistics,  assumes 
independence  of  up-crossings,  by  ignoring  ‘bandwidth’  [5],  an  effect  related  to 
the  concentration  of  energy  on  response  sample  paths  within  a  narrow  band  of 
frequencies. 

Threshold  crossing  statistics  can  be  approached  via  the  joint  pdf 
for  a  response  amplitude  process  and  its  first  derivative,  which  in  turn  can  be 
obtained  for  a  general  class  of  non-linear  systems  via  Markov  process  theory 
using  the  stationary  Fokker-Planck-Kolmogorov  (FPK)  equation  [2]  [10].  This 
equation  can  often  be  constructed  directly  from  the  equations  of  motion  for  a 
stochastic  dynamic  system,  provided  the  excitation  sources  can  be  modelled  as 
broad-band.  Although  few  exact  FPK  solutions  are  known,  in  the  past  30  years 
there  have  been  several  types  of  numerical  method  proposed  (references  [11- 
16]  are  just  some  examples  of  functional- type  solutions,  not  to  mention  finite 
difference  methods).  A  good  numerical  solution  opens-up  possibilities  for 
obtaining  accurate  tails  of  the  response  amplitude  pdf,  and  accurate  extreme- 
value  statistics,  thereby  avoiding  the  need  for  enormously  long  simulations. 
Whilst  numerical  FPK  solution  methods  obviously  differ  in  accuracy  and 
efficiency,  there  is  one  school  of  thought  to  suggest  that  finite  element 
structural  modelling,  followed  by  FE  solution  of  the  FPK  equation,  offers  an 
attractive  and  unified  approach  to  structural  reliability  assessment. 

But  the  accuracy  of  FEM-FPK  predictions  depend  on  the  use 
of  a  sufficient  number  of  elements  in  the  approximation,  and  to  achieve  a 
certain  level  of  accuracy,  this  is  affected  by  the  dimension  of  the  problem,  the 
strength  of  nonlinearity  in  the  model,  and  on  the  probability  levels  considered. 
Computer  storage  limitations  may  also  prevent  the  required  number  of  elements 
from  being  used.  Furthermore  stationary  FPK  solutions  contain  no  ‘bandwidth’ 
information  at  the  sample  level.  Therefore,  the  assumption  of  independent  up- 
crossings,  intrinsic  to  all  FPK  based  extreme- value  predictions,  must  be 


1598 


checked  using  an  alternative  such  as  Monte  Carlo  simulation.  But  to  justify 
such  lengthy  checking,  a  realistic  non-Hnear  model  is  appropriate,  for  example 
a  large  amphtude  model  for  clamped-clamped  beam  vibration  excited  by  broad¬ 
band  noise  [17]. 

In  this  paper,  a  SDOF  beam  model,  and  a  pair  of  non-hnear 
oscillators,  are  used  to  demonstrate  use  of  Langley’s  FEM-FPK  method.  In 
both  models,  the  parameters  are  known,  and  only  external  Gaussian  white- 
noise  excitations  are  assumed.  For  the  SDOF  (beam)  model,  the  rate  of 
convergence  of  the  response  amplitude  pdf  via  the  FEM,  is  demonstrated  as, 
more  nodes  are  used.  FEM-FPK  based  extreme  exceedance  probabilities  are 
then  compared  with  long  Monte  Carlo  simulations  to  show  convergence  with 
increasing  nodes,  and  to  confirm  the  iusensitivity  of  bandwidth  effects  at  the 
probabihty  levels  considered.  The  model  is  justified  at  the  pdf  level,  by  showing 
calibrated  FEM-FPK  predictions  and  experimental  measurements.  A  4D  FEM- 
FPK  version  is  then  applied  to  a  pair  of  non-linear  oscillators.  This  is  to 
establish  the  level  of  convergence  reached  in  the  response  amphtude  pdf,  with 
practical  storage  limitations  imposed  using  a  systematic  FEM  (global)  node 
numbering  system. 

2.  The  FPK  Equation  and  Stationary  Response  statistics 

When  the  force  vector  components  arising  in  a  structural  dynamic  model  can  be 
treated  as  broad-band  Gaussian  processes,  Markov  process  theory  allows  the 
(forward)  transition  probabihty  density  function  to  be  modehed  using  the 
Fokker-Planck  (FPK)  equation  (and  backward  transitions  via  the  Backward 
equation)  [2].  From  these  densities  important  response  statistics  can  be 
obtained  for  use  in  rehabihty  assessment.  Under  certain  conditions,  statistical 
properties  of  response  trajectories  cease  to  vary  with  time  aUowing  a  simpler 
(stationary)  FPK  equation  to  be  constructed.  The  starting  point  for 
constructing  the  stationary  FPK  [10]  is  to  express  the  system  of  (non-hnear) 
equations  of  motion: 

MX  +  F{X,X)  =  f{t)  (1) 

in  appropriate  state  space  fonn  as  follows: 

i  =  ^(^)  +  Aw(0  (2) 


where  z  represents  an  nxl  vector  Markov  response  process,  w(0  is  an 
assumed  nxl  vector  of  zero-mean  uncorrelated  Gaussian  white-noise 
processes,  whose  spectral  densities  are  unity,  scaled  by  a  constant  square 
matrix  A,  and  g  is  an  nxl  vector  of  (non-hnear)  system  functions.  The 

stationary  FPK  equation  associated  with  equation  (2),  can  be  obtained  by 


1599 


identifying  so  called  drift  and  diffusion  coefficients  [10]  and  by  focusing  on 
steady-state  trajectories.  This  leads  to  the  partial  differential  equation: 


(3) 


where  5  =  2;&4A^and  p(z)  is  the  stationary  joint  probability  density  function 
(jpdf)  associated  with  the  solution  trajectories  of  the  dynamic  model.  Boundary 
conditions  take  a  variety  of  forms  [2],  in  addition  to  the  normalisation  condition 
requiring  that  the  total  probability  must  sum  to  one.  Very  few  exact  solutions 
to  equation  (3)  are  known  which  is  one  important  reason  why  numerical 
methods  are  increasingly  used. 

2.1  Marginal  Densities  and  Extreme- Value  Statistics 

From  the  stationary  FPK  solution  p(z),  marginal  density  functions,  and 
corresponding  moments,  can  be  obtained  by  successive  integration.  The  joint 
bi-variate  pdf  for  a  response  and  its  first  derivative,  can  be  used  to  obtain  the 
mean  threshold  crossing  rate  [10].  If  this  is  accurate,  then  it  offers  an 
approximate  route  to  the  distribution  of  extreme-values  for  response  process 
Zi .  The  required  bivariate  density  function  is  in  general  obtained  by  integration 

as  follows: 


+<« 

and  from  equation  (4),  the  mean  up  crossing  rate  is  determined  from: 

4«> 

=  j  Zi+l/’(UT.Z.+l)*W  (5) 

0 

where  u^  refers  to  the  level  of  interest,  and  dz  corresponds  to  the  reduced 
state  space  in  which  variables  z,  and  z^+i  do  not  appear.  The  Poisson 
assumption  of  independent  upcrossings  [6]  gives  the  approximate  extreme- 
value  distribution  for  z,-  (t)  in  the  form: 

■Fm(Ut)  =  prob{M(T)  <  %}  =  (6) 

where  M  is  the  largest  value  of  response  process  z,-  (0  in  the  interval  T.  From 
this,  the  exceedance  probability  for  extreme-value  is  defined  as: 


1600 


P{M(T)  >  u^}=1-F(Ut) 


(7) 


3.  Finite  Element  Solution  of  the  Stationary  FPK  Equation 

There  have  been  several  finite  element  FPK  solution  methods  proposed  such  as 
the  methods  of  Langley[12],  Bergman[14],  Spencer  and  Bergman[16]  (also 
applied  to  the  Backward  equation).  These  methods  are  essentially  special 
applications  of  the  weighted  residual  (WRM)  Galerkin-type  approximation, 
using  shape  functions  defined  over  a  fimte,  rather  than  infimte  region.  The 
Finite  element  method,  for  example  proposed  by  Langley,  has  the  additional 
advantage  of  being  able  to  deal  efficiently  with  spatial  dependence  in  the  FPK 
equation  -  a  feature  which  does  not  normally  occur  in  general  structural 
analysis  problems.  We  do  not  propose  to  examine  these  methods  in  any  detail, 
except  to  give  a  brief  outline  of  Langley’s  method  in  the  usual  context  of  the 
WRM  appropriate  for  a  stationary  multi-dimensional  FPK  equation  since  this  is 
used  later. 

In  Langley’s  method  [12],  general  use  is  made  of  piece-wise 
shape  functions  to  approximate,  within  a  number  finite  elements,  both  the 
solution  P(z),  and  the  non-linear  appearing  in  equation  (3).  These 

elements  are  usually  rectangular  for  2D,  cuboid  for  3D  etc.,  and  the  total 
number  of  nodes  for  a  single  n-dimensional  element  is  m  =  2"  •  The  solution 
P{z)  over  the  entire  domain  is  approximated  in  terms  of  the  (unknown)  nodal 

values  of  the  pdf  as  follows: 


(8) 


Shape  functions  N,  (2)  are  chosen  to  take  unit  values  at  a  specified  node,  and 
zero  at  other  nodes  withm  a  particular  element.  For  nodes  which  do  not  fall 
within  a  particular  element,  the  shape  function  values  are  zero. 

In  trial  solution  (8),  the  usual  far-field  boundary  conditions  are 
modified  to  account  for  use  of  a  finite  region,  namely  that  p{z)  0  when 
(2)  — >  ±a  where  a  is  some  large  finite  value.  Substitution  of  trial  solution  (8) 
for  all  elements  over  the  entire  finite  region,  allows  the  FPK  equation  residual 
to  be  formed  as  follows: 


^  i=l  ;=1  CfZi  f=l  OZi 


(9) 


Equation  (9)  gives  a  measure  of  how  well  the  entire  solution  p{z)  satisfies 
equation  (2)  and  if  the  shape  functions  are  chosen  correctly,  one  hopes  that  the 


1601 


residual  will  reduce  uniformly  to  zero  as  the  number  of  elements  is  increased. 
Weighted  residuals  are  formed  as  follows: 


j  j  R{^w{£)dz  =  0 


(10) 


to  provide  the  basis  for  selecting  the  nodal  values  of  the  approximate  solution. 
The  weight  functions  w(z) ,  are  chosen  appropriately  to  be  the  same  nodal 
shape  functions  within  those  elements  in  which  a  particular  node  of 
interest  falls.  For  all  other  elements,  the  weight  functions  are  taken  to  be  zero. 
This  converts  the  weighted  residual  statement  (10)  to  a  finite  number  of 
summations  over  all  elements,  which  in  principle  generates  the  same  number  of 
(linear)  equations  as  unknown  nodal  probability  values.  This  (generally  sparse) 
system  of  linear  equations  is  in  fact  singular,  but  a  unique  solution  can  be 

obtained  by  imposing  the  normalisation  condition: 

+« 

lp(.l)dg  =  l  (11) 

To  make  best  use  of  finite  elements,  the  simplest  type  of  shape  function  is  used 
to  remove  much  of  the  burden  in  organising  the  FE  code.  The  simplest  choice 
is  indeed  a  piece-wise  linear  (Lagrange)  function,  where  n-dimensional  shape 
functions  can  be  constructed  using  appropriate  multiples  of  the  one-dimensional 
version.  However,  were  a  trial  solution  to  be  attempted  using  these  Lagrange- 
type  functions  directly  in  equation  (10),  the  FPK  equation  being  second  order, 
would  cause  the  first  term  to  vanish  since  linear  shape  functions  have  only  Cg 
continuity.  This  problem  was  overcome  [12]  by  integrating  equation  (10),  to 
generate  the  weak  form  of  weighted  residual  statement  for  the  FPK  equation: 


X  ^ J  ^  I  (2)p(s)^[w(z)](iz  =  0 


j=l 


i=l  R 


(12) 


Equation(12)  applied  over  aU  the  finite  elements  will  yield  a  system  of  linear 
equations  for  p.  which  can  be  obtained  uniquely  by  imposing  the  normalisation 
condition  (11).  As  mentioned,  far-field  boundary  conditions  are  applied  at  a 
finite,  rather  than  an  infinite  boundary,  determined  in  practice  using  equivalent 
linearization.  This  boundary  is  chosen  at  5  or  6  standard  deviations  in  aU 
directions  where  the  probability  density  outside  this  region  is  assumed  to  be 
zero.  The  non-linear  functions  g;(2)  appearing  in  equation  (3)  can  also,  for 
reasons  of  computational  efficiency,  be  approximated  within  a  particular 
element  using  the  same  shape  functions  as  follows: 

m 

g(z)=J,gm)  (13) 

/=i 


1602 


This  has  the  advantage  of  reducing  the  amount  of  pre-processing  required  to 
handle  the  spatial  variability  in  the  FPK  equation  and  is  therefore  useful  where 
parameter  or  model  variations  are  intended.  Once  p(z)  is  calculated  in  the 
above  manner,  the  extreme- value  exceedance  distribution  can  be  approximated 
from  equations  (4)-(7). 

A  storage  problem  however  arises  in  the  use  of  the  FEM,  since 
the  linear  coefficient  matrix  is  of  dimension  x  ,  where  N  is  the  number 
of  nodes.  This  obviously  becomes  excessive  for  anything  more  than  the  2- 
dimensional  problem.  Iterative  solution  techniques  [18]  can  be  used  to  solve 
linear  equations  without  need  to  store  the  coefficient  matrix.  But  these  methods 
are  suited  to  sparse  matrices  with  narrow  bandwidth,  such  as  arise  in  structural 
FEM  -  the  best  known  being  the  Frontal  solution  method.  Unfortunately  the 
benefits  of  using  space  saving  methods  in  the  FEM-FPK  are  to  some  extent 
lost,  as  it  can  be  shown  that  even  if  the  theoretical  bandwidth  is  no  more  than 
3^^ ,  the  practical  bandwidth  is  significantly  more  than  this  when  a  systematic 
nodal  numbering  scheme  is  used.  Therefore  for  large  n  (say  >  3),  unless  an 
extremely  sophisticated  global  node  numbering  system  is  used,  the  bandwidth 
grows  rapidly  with  the  node  number  and  the  advantages  of  space  saving 
methods  are  not  realised.  For  small  n  (e.g.  n=l  or  2)  storage  requirements  are 
not  so  critical,  so  a  good  solution  can  be  obtained  using  standard  sparse 
system  solution  techniques. 

4.  Application  to  SDOF  Model 

The  first  application  of  this  FEM-FPK  is  to  a  SDOF  oscillator  model: 

Z  +  2^a),Z  +  a,ZZ~  +  o:,z|z|  +  colZ  +  k^Z^  =  Aw{t)  (14) 

which  represents  a  system  with  both  hnear  and  non-linear  damping,  and  linear 
plus  cubic  stiffness.  The  excitation  w(0  is  a  unit  intensity  stationary  zero  mean 
Gaussian  white  noise  process,  scaled  to  required  level  by  parameter  A.  Some 
evidence  will  be  shown  shortly  to  confirm  this  as  a  realistic  vibration  model  for 
one  dimensional  large  amplitude  vibration  of  clamped-clamped  beam,  at  least 
in  the  tails  of  the  response  amplitude  marginal  probability  density  function  [17]. 
First  we  intend  to  show  convergence  of  predicted  FEM-FPK  pdf  solutions  with 
increasing  nodes;  and  then  extreme  exceedance  predictions  based  on  the  FEM- 
FPK  using  equations  (3)-(7),  in  both  cases  predictions  are  compared  with 
Monte  Carlo  simulations  -  this  includes  an  assessment  of  the  Poisson 
assumption  using  equation(5).  To  complete  the  section,  pdf  predictions  using  a 
calibrated  form  of  model  (14),  are  compared  with  experimental  measurements 
of  clamped-clamped  beam  vibration. 

Putting  oscillator  model  (14)  into  state  space  fonn,  gives  a  two- 
state  vector  Markov  model: 


1603 


(15) 


Z2 

-a^z2Zf  -a2Z2lz2l~0)Ui 


"1 

’0  0* 

“  0  ■ 

dt  -j- 

_0  A_ 

db{t)_ 

and  the  corresponding  stationary  FPK  equation  (3)  can  then  be  written  as: 


M^-^-^(zi/(z))  +  -^((2(?a)„Z2  +  «iZ2zf  +  +^3^1  )/(^))  -  ^ 

UZ2  ^Z\  0^2 

...  (16) 

subject  to  far-field  and  normalisation  boundary  conditions  as  described  in 
section  2.  The  parameter  values  used  for  the  model  are  : 
^  =  0.0, =  0M2,a,  =  0.015 ,  =  144.34,^  =  3021 ,  A  =  200 .  Application 

of  statistical  linearization  gives  =0.0138  and  simulation  -reveals  a 

eq 

bandwidth  parameter  value  £  =  0.97 .  Note  although  expKcit  parameter  values 
are  given  here,  only  damping  to  intensity  ratios  appear  in  the  FPK  solution, 
which  confirms  that  bandwidth  effects  cannot  be  accounted  for,  hence  the  need 
for  an  independent  check  on  the  accuracy  of  extreme-value  predictions. 


Figure  1. :  Response  amplitude  marginal  density  function  for  SDOF  model  via  FEM-FPK 
solutions  and  simulation. 

To  check  FEM-FPK  predictions,  conventional  time-domain 
simulations  are  used  here.  These  involve  three  stages  :  1)  construction  of 


1604 


excitation  sample  paths  ;  2)  numerical  integration,  using  a  standard  4th  order 
Runge-Kutta  scheme;  and  3)  post-processing,  involving  transient  removal  to 
create  a  number  of  output  sample  paths  for  estimation  of  marginal  response 
densities,  and  for  creation  of  sections  of  length  T,  for  use  in  estimation  of 
extreme  exceedance  probabilities.  A  truncated  Whitaker  filter  [19]  is  used  to 
allow  convergence  of  numerical  integration  using  white-noise  excitation 
samples  assembled  at  discrete  time  intervals  of  duration  At ,  giving  a  Nyquist 
bandwidth  f,,  =  .  In  the  use  of  the  Whitaker  filter  At  is  fixed,  aUowing 

interpolation  to  smaller  time  steps  At,  producing  rapid  reduction  of  the 
truncation  error. 

Figure  1.  shows  the  response  marginal  pdf  obtained  via  FEM- 
FPK  and  simulation.  Using  symmetry  over  1/4  region,  the  number  of  nodes  in 
the  FEM  is  increased  as  shown,  from  9  nodes  to  a  total  of  225.  This  number 
does  not  create  computational  or  storage  problems  using  a  systematic  FEM 
node  numbering  scheme,  and  convergence  has  clearly  occurred  by  225  nodes 

Figure  2.  shows  extreme  exceedance  probabilities  using  FEM- 
FPK  predictions  via  the  threshold  crossing  rate  equation  (5)  in  equations  (6) 
and  (7),  compared  with  very  long  Monte  Carlo  simulations.  Two  durations 
have  been  examined;  T=1  second,  as  shown  in  figure  2a.  ;  and  T=100  seconds 
in  figure  2b.  In  both  figures,  convergence  of  FEM-FPK  predictions  are 
compared  with  converging  simulation  results  for  reducing  At  in  the  Whitaker 
filter.  Note  that  the  simulations  seem  to  have  converged  for 
At  =  0.001  seconds  ,  and  the  FEM-FPK  predictions  seem  to  similarly  converge 
but  only  when  a  total  of  961  nodes  is  used.  Results  for  225  nodes,  which  are 
perfectly  adequate  for  marginal  density  predictions  as  shown  in  figure  1,  are 
clearly  not  accurate  for  extreme-value  statistics.  The  final  level  of  agreement 
between  simulation  and  FEM-FPK  predictions  would  suggest  that  the  Poisson 
assumption  does  indeed  hold  below  probabilities  of  10  ^  justifying  use  of 
equation(6). 

The  statistical  variability  m  extreme  exceedance  estimates  p  is 
predicted  using  the  ratio  cr.  /  p  =  l/ Np  ,  where  cr *  is  the  standard  deviation 
in  p  and  N  is  the  sample  size.  For  low  probabilities,  a  large  value  N=1000 
extreme-values  was  needed  for  each  duration,  giving  confident  estimates  of  p 
above  10'^.  The  CPU  time  needed  here  for  the  simulations  varied  between  30 
and  300  times  the  FEM-FPK  requirement. 

Figure  3  shows  FEM-FPK  predictions,  based  on  a  calibrated 
SDOF  model  (14),  via  the  parameter  estimation  method  [20],  compared  with 
direct  measurements  of  large  amplitude  beam  vibrations  [17].  This  experimental 
clamped-clamped  beam  rig  was  Im  in  length,  and  25mm  by  3mm  section,  with 
density  p  =  7850kg /m^  and  Young’s  Modulus  E  =  190GN/m^  Band- 
limited  white  noise  excitation  was  applied  using  a  shaker  positioned  near  one 


1605 


Figure  2. :  Extreme  Exceedance  probabilities  for  SDOF  model.  FEM-FPK  predictions 
compared  with  Monte  Carlo  simulations:  a)  T=1  second;  b)  T=100  seconds. 

- Simulated  Response  (delta  t=0.004  sec)  - Simulated  Response  (delta  t=0.002  sec) 

.  Simulated  Response  (delta  t=0.001  sec)  —A—  F.E  15x15=225  nodes 

RE  21x21=441  nodes  -X-  F.E  31x31=961  nodes 


end  of  the  beam,  the  central  ampHtiide  being  measured  with  an  accelerometer. 
Full  details  of  the  rig,  instrumentation,  and  data  processing  procedure  can  be 


1606 


0.5 


Figure  3. :  Response  amplitude  marginal  density  functions  for  beam  model.  Calibrated  SDOF 
model  showing  FEM-FPK  predictions  and  experimental  measurements:  a)  linear 
model;  b)  non-linear  model. 

found  in  [17].  For  this  beam,  linear  theory  gives  the  first  three  small  amplitude 
natural  frequencies  as  15Hz,  42Hz,  and  82Hz  respectively,  whereas  measured 
large  amplitude  responses  are  concentrated  around  25Hz  in  a  single-degree- 
of-freedom  type  motion  without  any  observed  responses  above  this  bandwidth. 
Figure  3a  shows  a  calibrated  linear  model,  in  contrast  to  figure  3b,  which 
compares  a  calibrated  version  of  equation  (14).  The  magnitudes  of  the  raw 
parameter  estimates  obtained  using  the  moment  method  [20]  are  given  as  : 
<J/A'  =5.09xl0■^  ajA^  =18.26xl0■^  =0.47x1 0"^  =132.56, 


1607 


and  =  2322 .  Note,  the  intensity  level  A  need  not  be  known  for  FEM-FPK 
predictions  at  the  same  excitation  level  as  the  measured  data.  This  is 
advantageous  since  excitation  may  in  general  be  difficult  to  measure. 


5.  Application  to  a  pair  of  non-linear  oscillators 

A  4D  FEM-FPK  version  can  be  applied  for  example  to  an  isolating  suspension 
system  [21],  which  in  general  represents  a  pair  of  coupled  non-linear  equations: 

X  +  od -h K^x  +  ‘¥yxO^  = 

0  -f- 156  +  G^O  +  G-^6^  -f*  y  6x^  —  ^2^2  (0 

By  setting  Zi=x,  Z2=0  and  Z3  =  x,  Z4  =  0  the  state  space  model  becomes: 


Z3  ^ 

ro 

0 

0 

0  ^ 

0  ^ 

-azs 

-K^Zi  -K^zl  -Jz^zl 

+ 

0 

A 

0 

A  2 

W;  (f) 

^3 

Z4 

0 

0 

0 

0 

0 

~  (JjZt  “^3^2  ^yZ'iZ^  J 

lo 

Ail 

0 

^2 ) 

where  A^  and  A2  are  white  noise  intensities;  Aj2and  A^^  are  cross-correlations, 
all  scaled  by  a  factor  In: .  The  FPK  equation  can  be  written  as  follows: 


2nA{  -f-  2nA,  + ;rA2  — {z^p)  - — {z^p) 


dzl  *“  dzsdz^  “  ^4 

^4 


6z^ 


+^[(a^3  +^1^1 + K.x  +n,zl)p]+-^[{i5z,  +  g,z2 + G2Z2  +rz,zl)p\  =  0 

C^3  OZ4 

...  (19) 

Equation  (19)  can  be  solved  exactly  in  specific  cases,  such  as  when: 
A  A 

Ai2  =  A21  =  0,—  =  -^  =  27  (a  nonzero  constant),  the  solution  is  then: 

a  p 


1  fl 


1. 


1 


1 


1 


p(^i,Z2,^»Z4)~C©qpj^  +^^Zi  ■{^Qz2+^Qz2'^~^y2:iZ2|  2^  ^3  2A 

...  (20) 

To  establish  the  finite  region  for  application  of  the  4D  FEM-FPK,  the 
equivalent  linear  fonns  of  equations  (17)  are: 


a 


x  +  ax  +  {K^  +  2>K^a\  +y(jl)x  =  A^a:^  {t) 
6  +  j5B  +  {G^  +3G3(Tg  +yal)B  =  A^w^_{t) 


(21) 


Solving  (21)  for  the  amplitude  and  velocity  mean-square  values  leads  to: 


1608 


<7!  = 


^  0  Ml 

and  = — - 


a 


(22)  &  (23) 


(^1  = 


- - r-  and  (jI 

l5(.G,+3G,<Ji+r<yl)  " 


,2  _  ^Ai 
a 


^  (24)  &  (25) 


To  demonstrate  this  4D  FEM-FPK,  the  parameter  values  were  set  to: 
if,  =  Gi  =  1.0  and  with  r  =  0.0and  A^=A^=\l4n.  Note, 

although  introducing  no  change  to  the  FEM-FPK  solution  method,  setting 
7  =  0.0  effectively  uncouples  equations  (17)  -  which  can  in  fact  then  be  solved 
with  a  2D  FPK  [10].  But  from  the  FEM  viewpoint  the  problem  can  still  be 


Figure  4. :  Response  amplitude  marginal  density  function  for  pair  of  nonlinear  oscillators. 

4D  FEM-FPK  predictions  compared  with  exact  solution. 

seen  as  if  it  were  4D,  the  advantage  being  that  exact  marginal  densities  can  be 
trivially  obtained  from  the  exact  jpdf.  Use  of  statistical  linearisation  [1],  gives 
(after  1 1  iterations)  converged  values  of  =  cr^  =  0.538  which  are  used, 
along  with  symmetry,  to  define  the  finite  FE  region: 
0<zi  <3.7,0<z,  <5.0,0<z3  <3.7,0<z4<5.0.  ^  made-up  of 

4D  equivalents  of  cuboid  elements,  each  with  16  nodes.  Figure  4  shows 
response  amplitude  marginal  density  predictions  obtained  with  this  4D  FEM- 


1609 


FPK  compared  with  the  exact  solution.  Only  two  cases  are  considered, 
namely  3*^  =  81  and  6^  =  1296  nodes  respectively.  This  second  case  involves 
solution  of  a  linear  system  of  equations  in  1296  unknowns.  But  since  a 
systematic  node  numbering  system  is  used,  the  bandwidth  of  the  coefficient 
matrix  is  very  much  larger  than  the  minimum  value  of  81.  Although  the 
comparison  for  the  second  case,  shows  good  agreement,  very  many  more  nodes 
would  clearly  be  needed  to  obtain  accurate  extreme- value  statistics.  This  would 
not  be  possible  with  the  present  approach  owing  to  computer  storage 
limitations. 


6.  Conclusions 

Accurate  stationary  response  amplitude  pdfs  are  shown  to  be  efficiently 
obtained  using  a  2D  FEM-FPK  approach  applied  to  a  realistic  non-linear  model 
for  SDOF  clamped-clamped  beam  vibration.  This  model  has  been  corroborated 
experimentally  at  the  marginal  density  level.  Application  of  the  FEM  to 
extreme-value  prediction,  via  threshold  crossing  statistics,  shows  good 
agreement  compared  with  simulation,  for  a  beam  type  model.  Moreover, 
application  of  a  4D  FEM-FPK  associated  with  a  pair  of  non-linear  oscillators, 
shows  reasonable  agreement  with  the  exact  solution  at  the  marginal  level.  But 
for  n  >  2,  use  of  a  systematic  FEM  node  numbering  scheme  is  not  suitable  for 
extreme- value  prediction  owing  to  the  large  bandwidth  of  the  FEM  coefficient 
matrices.  To  circumvent  this  problem  for  application  to  higher  dimensions  a 
sophisticated  global  node  numbering  scheme  is  needed  to  enable  space  saving 
linear  equation  solution  techniques  to  be  of  benefit. 

References 

1.  Roberts,  J.B.  and  Spanos,  P.D.,  Random  Vibration  and  Statistical 
Linearization,  Wiley,  Chichester,  1990. 

2.  Lin,  Y.K.  and  Cai,  G.Q.,  Probabilistic  Structural  Dynamics,  Mc-Graw- 
Hill.  1995. 

3.  Leadbetter,  M.R.,  Lindgren,  G.  and  Rootzen,  H.,  Extremes  and  related 
properties  of  random  sequences  and  processes.  New  York  :  Springer 

-  Verlag  1983. 

4.  Winterstein,  S.R.  and  Ness,  O.B.,  Hennite  moment  analysis  of  non-linear 
random  vibration,  in  Computational  Mechanics  of  Probabilistic  and 
Reliability  Analysis,  Lausanne;  Ehne  Press,  1989, Chapter  21, pp.  452- 
478. 

5.  Naess,  A.,  Approximate  first-passage  and  extremes  of  narrow-band 
Gaussian  and  Non-Gaussian  random  vibration.  Journal  of  Sound  and 
Vibration,  1990,  138,  pp.  365-380. 

6.  Naess,  A.  Galeazzi,  F.  and  Dogliani,  M.,  Extreme  Response  predictions 
of  non-linear  compliant  offshore  structures  by  stochastic  linearization  . 
Applied  Ocean  Research,  1992,14,  pp.  71-81. 


1610 


7.  Winterstein,  S.  R.  and  Torhaug,  R.,  Extreme  response  of  Jack-up 
structures  from  limited  Time-Domain  Simulation.  Proceedings  of  the  12th 
International  Conference  of  the  0 MAE- ASMS,  1993,  Vol  2,  251-258. 

8.  Dunne,  J.F.,  An  optimal  control  approach  to  extreme  local  maxima  for 
stochastic  Duffing-type  oscillators.  Journal  of  Sound  and  Vibration,  1996, 
193(3),  pp.  597-629. 

9.  Naess,  A.,  On  a  rational  approach  to  extreme  value  analysis:  Technical 
nolQ.  Applied  Ocean  Research,  1984  ,6(3),  pp.  173-174. 

10.  Soong,  T.T. ,  Random  Differential  Equations  in  Science  and  Engineering, 
Academic  Press  -  New  York,  1973. 

1 1 .  Bhandari,  R.G.  and  Sherrer,  R.E., ,  Random  vibration  in  discrete  non 
-linear  dynamic  systems.  Journal  of  Mechanical  Engineering  Science, 

1968,  10,  pp.  168-174. 

12.  Langley,  R.S.,  A  finite  element  method  for  the  statistics  of  non-linear 
random  vibration.  Journal  of  Sound  and  Vibration,  1985,  101(1),  pp.  41 
-54. 

13.  Soize,  C.,  Steady  state  solution  of  Fokker-Planck  equation  for  higher 
dimensions.  Probabilistic  Engineering  Mechanics,  1989,  3,  pp.  196-206. 

14.  Bergman,  L.A.,  Numerical  solution  of  the  first  passage  problem  in 
stochastic  structural  dynamics,  in  Computational  Mechanics  of 
Probabilistic  and  Reliability  Analysis;  Lausanne;  Elme  Press  1989,  pp. 
479-508. 

15.  Kunert,  A.,  Efficient  numerical  solution  of  multidimensional  Fokker- 
Planck  equations  associated  with  chaotic  and  non-linear  random 
vibrations.  Vibration  Analysis  -  Analytical  and  Computational,  ASME 
DE-57, 199 l,pp.  57-60. 

16.  Spencer,  Jr.  B.F.  and  Bergman,  L.A.,  On  the  numerical  solution  of  the 
Fokker  Planck  equation  for  nonlinear  stochastic  system.  Non-linear 
Dynamics,  1993,  Vol  4  ,  pp.  357-372. 

17.  Ghanbari,  M.,  Extreme  response  prediction  for  random  vibration  of  a 
clamped- clamped  beam,  1996,  D.Phil.  Dissertation  -  University  of 
Sussex. 

18.  Press  W.H.,  Flannery,  B.  P.,  Teukolsky,  S.  A.  and  Vetterling,  W.T., 
Numerical  Recipes  -  the  art  of  scientific  computing,  Cambridge 
University  Press,  1989. 

19.  Oppenheim,  B.W.  and  Wilson,  P.  A.,  Continuous  digital  simulation  of 
the  second  order  slowly  varying  drift  force.  Journal  of  Ship  Research, 
1980,  24(3),  pp.  181-189. 

20.  Roberts,  J.B.,  Dunne,  J.F.  and  Debonos,  A.,  Parameter  estimation  for 
Randomly  excited  non-linear  systems,  in  lUTAM  symposium  on  Advances 
in  Non-linear  Stochastic  Mechanics,  Kluwer,  1996,  pp.  361-372. 

21.  Ariaratnam,  S.T.,  Random  vibrations  of  non-linear  suspensions.  Journal  of 
Mechanical  Engineering  Science,  1960,  2,  pp.  195-201. 


1611 


1612 


RANDOM  VIBRATION  II 


SIMULATION  OF  NONLINEAR  RANDOM  VIBRATIONS 
USING  ARTIFICIAL  NEURAL  NETWORKS 


Thomas  L.  Paez* 

Susan  Tucker 
Chris  O’ Gorman 

Sandia  National  Laboratories 
Albuquerque,  New  Mexico,  USA 


Abstract 

The  simulation  of  mechanical  system  random  vibrations  is  important  in 
structural  dynamics,  but  it  is  particularly  difficult  when  the  system  under 
consideration  is  nonlinear.  Artificial  neural  networks  provide  a  useful  tool  for 
the  modeling  of  nonlinear  systems,  however,  such  modeling  may  be  inefficient 
or  insufficiently  accurate  when  the  system  under  consideration  is  complex. 
This  paper  shows  that  there  are  several  transformations  that  can  be  used  to 
uncouple  and  simplify  the  components  of  motion  of  a  complex  nonline^ 
system,  thereby  making  its  modeling  and  random  vibration  simulation,  via 
component  modeling  with  artificial  neural  networks,  a  much  simpler  problem. 
A  numerical  example  is  presented. 

Introduction 

Structural  system  random  vibration  simulations  are  required  in  a  wide 
variety  of  applications.  Development  of  techniques  that  can  generate  such 
simulations  accurately  and  efficiently  is  important,  particularly  in  frameworks 
where  numerous  simulations  are  required,  frameworks  tike  Monte  Carlo 
analysis.  In  practically  all  situations  where  the  excitation  is  Gaussian  and  the 
system  under  consideration  is  nonlinear,  the  responses  will  be  nonlinear  and 
non-Gaussian,  and  it  is  important  that  simulations  preserve  the  characteristics 
of  the  response  as  accurately  as  possible. 

Artificial  neural  networks  (ANNs)  have  been  applied  to  the  autoregressive 
modeling  of  nonlinear  system  random  vibrations.  Investigations  have  shown 
that  nonlinear  structures  can  be  modeled  with  ANNs,  at  least  in  the  case  of 
simple  systems.  (See,  for  example,  Yamamoto,  [15].)  In  principle, 
complicated  systems  can  also  be  mc^eled  using  ANNs.  This  can  be  done 
directly  (i.e.,  without  any  substantial  transformation  of  the  input  or  output 
data)  using  many  types  of  ANNs.  As  the  complexity  of  the  system  increases, 
an  ANN  that  can  naturally  and  efficiently  accommodate  a  large  number  of 
inputs  must  be  used  for  system  simulation.  When  a  mechanical  system  is 
modeled  using  an  autoregressive  ANN  to  directly  simulate  motions  at  a  large 

*  This  work  was  supported  by  the  United  States  Department  of  Energy  under 
contract  No.  DE-AC04-94AL85000.  Sandia  is  a  multiprogram  laboratory 
operated  by  Sandia  Corporation,  a  Lockheed  Martin  Company,  for  the  United 
States  Department  of  Energy. 


1613 


number  of  degrees  of  freedom,  a  very  large  number  of  exemplars  of  motion 
will  be  required  to  train  the  ANN  to  accurately  represent  the  system.  The 
reason  is  that  it  takes  a  large  number  of  exemplars  to  adequately  populate  a 
high  dimensional  input  space. 

This  paper  shows  how  the  ANN  modeling  of  nonlinear  structures  can  be 
made  more  efficient  and  accurate  when  using  data  measured  during 
experimental,  stationary  random  vibration.  There  are  a  number  of  operations 
that  can  be  performed  on  the  data  to  accomplish  these  goals.  Among  tiiese  are: 
(1)  principal  component  analysis,  (2)  localized  modal  filtering,  (3)  elimination 
of  statistically  dependent  components  of  motion,  and  (4)  transformation  of  the 
components  of  motion  to  statistically  independent,  standard  normal  random 
signals.  These  operations  are  briefly  describe  in  the  following  sections,  along 
with  the  modeling  of  the  components  with  two  types  of  ANN  -  the  feed 
forward  back  propagation  network  (BPN)  and  the  connectionist  normalized 
linear  spline  (CNLS)  network.  An  example  is  included  to  assess  the  random 
vibration  simulation  capabilities  of  the  ANNs.  The  accuracy  of  the  simulations 
is  evaluated  in  terms  of  spectral  and  probabilistic  measures. 

Data  Reduction 

It  is  important  to  reduce  the  dimensions  of  motion  of  a  complex  system  for 
the  reason  listed  in  the  introduction,  i.e.,  the  amount  of  data  required  to  train  a 
very  complex  system  directly  is  great.  Further,  because  the  CNLS  net  is  a  local 
approximation  network,  it  is  important  to  minimize  the  number  of  network 
inputs.  The  reason  is  tiiat  network  size  grows  rapidly  with  the  number  of 
inputs.  To  limit  the  complexity  of  the  input/ouiput  mappings  required  to  model 
a  complex  system,  the  system  motions  can  be  decomposed  into  simple 
components.  In  general,  the  ANN  modeling  of  physical  systems  can  often  be 
made  more  efficient  and  accurate  by  preprocessing  the  training  data  using  ^y 
of  a  number  of  simplifying  transformations.  Among  these  are:  (1)  principi 
component  analysis,  (2)  localized  modal  filtering,  (3)  elimination  of  statistically 
dependent  components  of  motion,  and  (4)  transformation  of  the  components  of 
motion  to  statistically  independent,  standard  normal  random  signals.  The  hopes 
in  using  these  transformations  are  that  the  ANN  required  to  model  a  component 
of  behavior  will  be  simpler  than  a  model  for  the  entire  system,  and  that  a 
simpler  model  will  be  easier  to  train.  Exactly  how  these  transformations  fit  into 
the  response  simulation  framework  will  be  discussed  more  later,  the  overall 
framework  is  described  in  Figure  4.  These  operations  are  briefly  described  in 
the  following  subsections. 

Principal  Component  Analysis  -  SVD 

Principal  component  analysis  of  complex  structural  system  motions  is 
aimed  at  decomposing  the  motions  into  their  essential  constituent  parts.  A 
special  example  of  this  is  the  modal  decomposition  of  linear  systems,  and 
analogous  decompositions  can  be  defined  for  nonlinear  systems  using,  for 
example,  singular  value  decomposition  (SVD),  or  a  principal  component 
analysis  ANN. 


1614 


SVD  is  described  in  detail,  for  example,  in  Golub  and  Van  Loan  [4].  It  can 
be  used  to  decompose  linear  or  nonlinear  structural  motions  in  the  following 
way.  Let  AT  be  an  nxN  matrix  representing  the  motion  of  a  structural  system 
at  N  transducer  locations  and  at  n  consecutive  times.  The  form  of  the  SVD  is 

X  =  UWV^  =  uwv^  (1) 

V  is  an  V  X  V  matrix;  its  columns  describe  the  characteristic  shapes  present  in 
the  rows  of  AT.  TV  is  an  V  x  V  diagonal  matrix  whose  nonnegative  elements 
characterize  the  amplitudes  of  the  corresponding  shapes  in  V.  The  elements  in 
TV  are  normally  arranged  in  descending  order.  Its  largest  elements  correspond 
to  the  most  important  components  in  the  representation,  is  an  nxN  matrix; 
its  columns  are  filtered  versions  of  the  motions  represented  by  the  columns  in 
X.  The  reason  that  U  is  said  to  be  a  filtered  form  of  the  motions  in  X  is  that 
the  columns  of  both  V  and  U  are  orthonormal  with  respect  to  themselves. 
Therefore, 


U  =  XVW~^ 


(2) 


and  T^TV  ^  serves  as  a  filtering  coefficient.  Because  some  of  the  elements  of  TV 
may  be  zero  or  nearly  zero,  indicating  components  that  do  not  contribute 

substantially  to  the  characterization  of  X,  the  elements  of  TV  ^  are  taken  as  the 
inverses  of  the  diagonal  elements  in  TV  that  are  greater  than  a  cutoff  level;  zero 

or  near  zero  elements  in  TV  are  replaced  with  zeros  in  TV”V  The  approximate 
equality  on  the  right  side  of  Eq.  (1)  indicates  that  some  components  of  the 
representation  can  be  zeroed,  and  still  maintain  a  good  approximation  to  A" .  In 
the  experimental  framework,  the  comjwnents  of  TV  whose  ratio  to  the 
maximum  value  is  lower  than  the  experiment  noise- to-signal  ratio  are  set  to 
zero.  The  matrices  «,  w,  and  v  are  the  matrices  I/,  TV,  and  V  with 
components  removed. 

The  columns  of  the  matrix  u  are  the  principal  components  of  the 
representation.  It  is  the  evolution  of  the  system  represented  in  the  columns  of  u 
which  we  seek  to  simulate.  Once  models  are  established  to  simulate  system 
response  through  simulation  of  the  columns  of  the  models  can  be  used 
along  with  system  initial  conditions  to  predict  structural  response.  The 
predicted  response  can  be  used,  along  with  Eq.  (1),  to  synthesize  response 
predictions  in  the  original  measurement  space. 

Principal  Component  Analysis  -  ANN 

ANNs  can  be  used  in  a  number  of  application  frameworks,  and  one  of  the 
focuses  of  this  paper  is  to  show  how  components  of  a  complex  system  motion 
can  be  modeled  with  ANNs.  However,  a  particular  ANN  can  also  be  used  as  a 
means  for  decomposing  complex  system  motions  into  simpler  components. 
This  ANN  is  the  principal  component  ANN  (PCANN).  (Baldi  and  Hornik, 
[2])  The  PCANN  is  simply  a  multi-layer  network  of  perceptrons  like  the 
standard  BPN  (Freeman  and  Skapura,  [3]),  but  it  has  a  particular  geometry, 


1615 


shown  in  Figure  1.  Let  (xy)  be  a  row  vector  of  N  elements,  the  jth  row  of  data 

in  the  matrix  X  defined  above.  Then  the  collection  of  all  the  rows  of  AT  provide 
n  exemplars  -  both  input  and  output  -  for  training  the  PCAIW  in  Figure  1. 
Some  important  features  of  the  ANN  in  Figure  1  are  that  (1)  it  is  a  BPN  with 
one  hidden  layer,  and  (2)  die  number  of  neurons,  R,  in  the  hidden  layer  is 
smaller  than  N,  ±e  number  of  columns  in  the  measurement  matrix.  The  idea 
behind  the  PCANN  is  that  it  compresses  the  information  in  the  input  layer  into 
the  information  present  in  the  hidden  layer,  then  uses  this  information  to 
reconstruct,  as  accurately  as  possible,  the  original  signal  on  the  output  layer. 
To  obtain  optimal  effect  from  the  PCANN,  sigmoidal  activation  functions 
would  normally  be  used  in  the  hidden  and  output  layer  neurons.  However, 
when  linear  activation  functions  are  used  in  tiie  hidden  and  output  layer 
neurons,  the  ANN  weights  are  related  to  the  components  of  the  SVD,  Eq.  (1). 


Figure  1.  Geometry  of  the  principal  component  artificial  neural  network. 

Let  the  jth  row  of  m  be  the  hidden  layer  outputs,  in  Figure  1.  The 

columns  of  u  are  the  principal  components  of  the  PCANN  representation.  It  is 
the  evolution  of  the  system  represented  in  the  columns  of  u  which  we  seek  to 
simulate.  Once  models  are  established  to  simulate  system  response  through 
simulation  of  the  columns  of  ii,  the  models  can  be  used  along  with  system 
initial  conditions  to  predict  stmctural  response.  The  predicted  response  can  be 
used,  along  with  the  portion  of  the  PCANN  to  the  right  of  the  hidden  layer,  to 
synthesize  response  predictions  in  the  original  measurement  space. 

Normally,  only  one  of  the  principal  component  analyses  described  in  this 
and  the  previous  sections  would  be  applied  to  the  data. 

Modal  Filtering 

Modal  decomposition  of  complex  system  motions  is  often  used  for  the 
simplification  of  mechanical  system  response  when  the  model  for  the  system  is 
assumed  linear  or  approximately  linear.  In  fact  though,  data-based  modal 


1616 


decomposition  can  be  used  on  any  collection  of  data;  its  purpose  is  to  break  the 
into  simple  narrowband  components,  thereby  simplifying  system 
characterization,  and  perhaps  simulation.  When  such  a  decomposition  is  used 
on  nonlinear  system  or  general  system  data,  it  is  often  referred  to  as  modal 
filtering. 

Modal  filtering  of  measured  data  can  be  performed  in  one  of  two 
frameworks.  First,  a  single  modal  analysis  can  be  used  to  filter  measured  da^ 
into  its  component  parts.  The  problem  with  this  is  that  when  a  system  is 
nonlinear,  its  characteristics  can  change  with  response  magnitude,  and  the 
number  of  principal  components  (modes)  also  changes.  It  is  difficult  for  a 
single  model  to  accurately  capture  such  changes.  The  second  alternative  is  to 
specify  multiple  modal  filters,  and  use  each  one  m  a  particular  range  of 
response  amplitudes.  A  very  simple  realization  of  this  type  of  analysis  would 
involve  the  use  of  two  modal  models.  One  would  be  applied  to  data  below  a 
particular  threshold,  and  the  other  would  be  applied  to  data  above  the 
threshold.  A  more  complicated  application  creates  a  linear  modal  model  at  each 
step  in  a  system  analysis.  Such  a  model  is  described  in  Hunter  [7]. 

The  form  of  a  modal  filter  is  similar  to  the  SVD,  but  the  means  for 
obtaining  the  filter  factors  is  much  different.  Meirovitch  [9]  describes  the 
theoretical  operations  included  in  ±e  definition  and  use  of  a  modal  filter. 
AUemang  and  Brown  [1]  outline  practical  means  for  performing  data-based 
modal  analysis.  The  form  of  the  modal  filter  can  be  expressed  as 

X  =  US<I>^  =  us^'^  (3) 

where  X  is  the  same  mechanical  system  motion  representation  as  above.  The 
columns  of  (p  represent  the  characteristic  shapes  of  the  system,  and  the  diagonal 
matrix  S  contains  normalizing  factors.  When  X  comes  from  a  line^  system  the 
columns  of  U  are  the  linear  modal  components  of  the  system  motion.  As  with 
the  SVD,  the  approximate  equality  on  the  right  indicates  that  some  components 
of  the  representation  can  be  eliminated,  and  still  maintain  a  good  approximation 
to  X.  It  is  the  evolution  of  the  columns  in  u  (a  reduced  form  of  U)  which  we 
seek  to  simulate.  Once  models  are  established  to  simulate  system  response 
through  simulation  of  the  columns  of  m,  the  models  can  be  used  along  with 
system  initial  conditions  to  predict  structural  response.  The  predicted  response 
can  be  used,  along  with  Eq.  (3),  to  synthesize  response  predictions  in  the 
original  measurement  space. 

When  multiple  decompositions  are  used  to  filter  the  motions  of  a  complex 
system,  then  multiple  expressions  like  Eq.  (3)  are  used  to  obtain  modal 
components. 

Elimination  of  Statistically  Dependent  Components 

Although  some  of  the  principal  component  analyses  of  the  previous 
sections  may  produce  orthogonal  components,  some  of  the  components  may 
be  completely  or  highly  statistically  dependent  upon  others.  For  example,  a 
structure  may  have  two  modes  with  nearly  the  same  modal  frequency.  One 


1617 


modal  motion  may  be  nearly  a  sine  wave,  and  the  other  may  be  nearly  a  cosine. 
The  motions  are  practically  orthogonal,  but  they  are  still  statistically  dependent. 
Statistical  independence  of  sources  implies  orthogonality,  but  orthogonality 
does  not  necessarily  imply  statistical  independence.  For  the  sake  of  efficiency, 
we  seek  to  eliminate  statistically  dependent  components  from  the  set  to  be 
modeled,  then  reintroduce  these  components  during  physical  system 
simulation.  In  this  way,  ANN  modeling  of  structural  behavior  is  simplifi^. 


When  a  dependency  exists,  it  can  be  characterized  using  the  conditional 
expected  value  of  the  variables  in  one  column  of  u  given  values  in  another 
column  of  m.  This  requires  approximation  of  a  joint  probability  density 
function  (pdf)  of  the  data,  and  this  can  be  obtained  using  the  kernel  density 
estimator.  (See  Silverman,  [13].)  The  pdf  approximation  is  known  as  the 

kernel  density  estimator  (kde).  Let  (wj),  j  = denote  the  row  vectors  of 
the  matrix  u.  The  kde  of  the  random  source  u  is  given  by 


/«(«)  = 


—  oo  <  M  <  oo 


(4) 


where  a  is  an  N  x  1  variate  vector,  K{.)  is  a  kernel  function,  and  is  a 
window  width  parameter  of  the  kernel  function.  The  kernel  function  can  be  my 
standard  probability  density  function,  and  often  the  pdf  of  a  multivariate 
standard  normal  random  vector  is  used.  That  is 


where  a:  is  the  N  x  1  variate  vector.  Using  the  kde  in  Eq.  (4),  the  estimator  for 
the  conditional  pdf  of  elements  in  one  column  of  u  given  the  values  in  another 
column  of  u  can  be  obtained  using  the  standard  formulas.  (See,  for  example, 
Papoulis  [11].) 

A  statistical  dependency  between  two  columns  of  u  can  be  detected  by 
forming  the  bivariate  pdf  estimator  of  the  random  source  of  the  two  columns 
using  the  kde  with  data  from  the  columns  in  question,  then  evaluating  and 
plotting  the  conditional  expected  value  of  one  variable,  given  a  range  of  values 
of  the  other  variable.  At  each  point  where  the  conditional  expected  value  is 
evaluated,  the  conditional  variance  can  also  be  evaluated.  The  conditional 
expected  value  and  variance  can  be  evaluated  for  situations  in  which  the  data  in 
the  two  columns  of  u  are  lagged  with  respect  to  one  another.  If  a  lag  is  found 
where  the  conditional  variance  is  uniformly  small,  i.e.,  small  at  all  locations 
defined  by  the  conditioning  variable,  then  a  statistical  dependency  has  been 
detected,  and  the  functional  form  of  the  dependency  is  defined  by  the 
conditional  expected  value.  The  dependent  variable  can  be  eliminated  from 
modeling  consideration.  When  modeling  has  been  completed  and  it  is 
necessary  to  restore  the  eliminated  component,  this  can  be  accomplished  using 
the  conditional  expected  value  develops  here. 


1618 


The  effect  of  eliminating  components  of  motion  that  are  completely 
dependent  on  other  components  is  to  eliminate  some  columns  in  the  matrix  u. 
Denote  the  reduced  matrix  Uj.  \  our  objective  is  to  model  the  evolution  of  the 
columns  of  Uj.  with  an  ANN. 

Rosenblatt  Transform 

The  previous  step  produces  a  description  of  the  motion  of  a  complex 
structure  in  terms  of  a  set  of  components,  none  of  which  is  completely 
statistically  dependent  on  others.  We  can  further  transform  the  components,  tiie 
columns  of  into  signals  that  are  statistically  independent  with  Gaussian 
distributions.  The  transformation  that  accomplishes  this  is  the  Rosenblatt 
transform.  (See  Rosenblatt,  [12].)  The  Rosenblatt  transform  has  the  following 
form. 


Z2  =  ^”'(^“2tel(“2'«l)) 


where  the  z^-,  i  =  1,...,N,  are  uncorrelated,  standard  normal  random  variables, 
0(.)  is  the  cumulative  distribution  function  (cdf)  of  a  standard  normal  random 

variable,  is  its  inverse,  and  the  F(.)  are  the  estimated  marginal  and 

conditional  cdf’s  of  the  random  variables  that  are  the  sources  of  the  colunms  of 
u.  These  approximate  cdf’s  can  be  obtained  by  integrating  Eq.  (4),  and  this  can 
be  accomplished  directly  when  the  kernel  used  in  Eq.  (4)  is  Eq.  (5). 

The  Rosenblatt  transformation  is  uniquely  invertible  because  the  exact  and 
approximate  cdfs  used  in  Eq.  (6)  are  monotone  increasing.  The  ckta  in  the 
matrix  w  can  be  transformed  to  the  standard  normal  space  by  using  it  in  Eq.  (6) 

in  place  of  the  a’s.  The  matrix  z  is  composed  of  the  elements  z/,  /  =  1,...,A^ , 
and  is  the  same  size  as  the  matrix  with  the  same  number  of  nonzero 
columns. 

It  is  the  evolution  of  the  values  in  these  columns  that  is  to  be  simulated  with 
ANNs.  Because  the  columns  in  z  are  statistically  independent,  we  need  only  to 
create  ANN  models  for  signals  in  individui  columns.  Once  models  are 
established  to  simulate  system  response  through  sirnulation  of  the  columns  of 
z,  the  models  can  be  used  along  with  system  initial  conditions  to  predict 
structural  response.  The  predicted  response  can  be  used,  along  with  the  inverse 
form  of  Eqs.  (6),  to  synthesize  response  predictions  in  the  original 
measurement  space. 


1619 


Modeling  of  Component  Motion  with  ANNs 

Our  ultimate  objective  is  to  simulate  complex  system  motion,  and  we  aim  to 
do  this  by  simulating  the  components  of  system  motion  obtained  using  the 
decompositions  and  transformations  described  above.  Many  ANNs  are  suitable 
for  this  task.  The  two  that  we  consider  in  this  paper  are  the  feed  forward  back 
propagation  network  (BPN)  and  the  connectionist  normalized  linear  spline 
(CNLS)  network.  The  BPN  is  the  most  widely  used  ANN  and  it  is  described 
in  detail  in  many  texts  and  papers,  for  example  Freeman  and  Skapura  [3],  and 
Haykin  [5].  The  BPN  is  very  general  in  the  sense  Aat  it  can  approximate 
mappings  of  relatively  low  or  very  high  input  dimension.  It  has  b^n  shown 
that,  given  sufficient  training  data,  a  BPN  with  at  least  one  hidden  layer  and 
sufficient  neurons  can  approximate  a  mapping  to  arbitrary  accuracy  (Homik, 
Stinchcombe,  and  White,  [6]). 

The  CNLS  network  is  an  extension  of  the  radial  basis  function  neural 
network  (Moody  and  Darken,  [10]).  It  is  described  in  detail  in  Jones,  et.al., 
[8].  It  is  designed  to  approximate  a  functional  mapping  by  superirriposing  the 
effects  of  basis  functions  that  approximate  the  mapping  in  local  regions  of  the 
input  space.  Because  it  is  a  local  approximation  neural  network,  we  cannot  use 
the  CNLS  network  to  accurately  approximate  mappings  involving  a  large 
number  of  inputs.  The  CNLS  network  has  not  been  widely  used  for  the 
simulation  of  oscillatory  system  behavior. 

To  simulate  a  column  in  z  using  either  of  the  ANNs  described  above,  we 
configure  the  net  in  an  autoregressive  framework.  This  configuration  uses  as 
inputs  previous  response  values  and  the  independent  excitation,  and  yields  on 
output,  the  current  response.  Figure  2  shows  such  an  application 
schematically.  The  quantity  zji  denotes  an  element  in  the  ith  column  of  z  at  the 

jth  time  index.  The  quantity  qj  denotes  the  excitation  at  the  jth  time  index.  Lj 

denotes  a  lag  index.  There  are  m  system  response  input  terras;  there  are  M+2 
excitation  terms  The  configuration  shown  in  Figure  2  implies  our  belief  that 
there  is  a  mapping 

^j+U  ~ 

and  that  the  ANNs  can  identify  that  mapping.  The  subscript  i  on  the  function 
g(.)  indicates  that  the  functional  mapping  varies  from  one  column  of  z  to  the 
next,  and  a  different  ANN  models  each  mapping.  It  is  normally  anticipated  that 

the  time  increment,  At,  separating  system  motion  measurements  that  are  the 
rows  of  the  matrix  X  is  small  relative  to  the  period  of  motion  of  the  highest 
frequency  component  we  intend  to  simulate. 

We  seek  to  train  both  types  of  ANN  to  model  the  behavior  of  the 
oscillations  represented  in  the  columns  of  z.  One  ANN  of  each  type  (BPN  and 
CNLS  net)  is  used  to  model  each  column  of  z.  The  inputs  to  the  ANN  are 
current  and  lagged  (past)  values  of  the  transformed  response  and  the  one-step- 
into-the-future  value,  the  current  value,  and  lagged  values  of  the  excitation. 
The  ANN  output  is  the  transformed  response  one  step  in  the  future.  Both  the 


1620 


ANNs  are  trained  using  the  scheme  described  in  Figure  3.  The  ANN  inputs 
are  transformed  using  a  feed  forward  operation.  The  ANN  output  is  compared 
to  the  desired  output,  and  the  error  is  used  to  modify  the  ANN  parameters.^  'Die 
BPN  uses  a  back  propagation  and  gradient  descent  scheme  in  each  trainmg 
step.  The  CNLS  network  uses  least  mean  square  (LMS)  plus  random  sampling 
scheme  to  identify  its  parameters.  The  desired  effect  of  training  in  both  types  of 
ANN  is  to  modify  the  parameters  of  the  network  to  diminish  the  error  of 
representation  of  the  input/output  mapping. 


^J-k4 


^i-T  i 

J 

qj+l 

Qj-^ 


A 

N 

N 


Figure  2.  Schematic  of  ANN  in  autoregressive  configuration. 


Figure  3.  Schematic  describing  training  sequence  for  ANNs. 

Summary 

Figure  4  summarizes  the  decomposition,  simulation,  and  modeling  of 
structural  motion  described  in  the  previous  sections.  The  principal  component 
analysis  in  the  second  box  in  the  top  row  refers  to  one  of  the  following:  SVD, 
PCANN,  or  modal  filtering.  The  synthesis  in  the  fourth  box  in  the  second  row 
refers  to  the  corresponding  inverse  operation  -  Eq.  (1),  the  right  half  of  Figure 
1,  or  Eq.  (3). 


1621 


In  the  example  that  follows,  one  form  of  principal  component  analysis  will 
be  combined  with  ANN  simulations  to  model  a  nonlinear  structure’s  random 
vibrations. 


Decomposition  and  Modeling  Operations 


Simulation  of  Response 


Figure  4.  Summary  of  operations  in  system  simulation. 

Numerical  Example 


We  simulate  in  this  example  the  motion  of  a  simple,  nonlinear  10  degree- 
of-freedom  system  excited  with  a  Gaussian  white  noise.  Figure  5  shows  a 
schematic  of  the  system.  The  damping  connecting  the  masses  is  linear  viscous. 
The  springs  have  a  restoring  force  that  is  a  tangent  function.  The  system 
physical  parameters  are  summarized  in  Table  1.  Training  data  for  the  neural 
networks  were  generated  by  computing  response  over  8192  time  steps  (box 
number  one  on  the  top  line  in  Figure  4);  excitation  and  responses  at  ten 
locations  were  recorded.  The  time  increment  between  response  realizations  is 
0.04  second.  Figure  6  shows  the  displacement  response  at  the  10th  mass.  This 
is  the  location  where  the  simulation-to-experiment  comparisons  are  made  in  the 
present  example,  and  where  the  simulation  yielded  the  poorest  match  to  the 
experimental  results. 

The  responses  were  placed  in  a  matrix  X  as  referred  to  in  the  previous 
sections,  and  its  SVD  was  computed  (box  number  two  on  the  top  line  of 
Figure  4).  The  singular  values  of  the  response  indicate  that  accuracy  of  about 
89%  should  be  achieved  by  simulating  the  system  response  with  its  first  four 
components.  The  four  components  were  not  strongly  statistically  dependent, 
so  none  was  removed.  The  kde  of  the  four  components  indicate  that  none  is 
highly  non-Gaussian,  therefore,  the  Rosenblatt  transform  was  not  used.  The 
first  four  components  of  the  response  were  modeled  with  both  BPN  and 
CNLS  nets  (box  number  five  on  the  top  line  in  Figure  4). 

The  entire  system  was  tested  in  autoregressive  operation,  as  described  in 
Figure  2,  using  data  generated  over  1000  steps  of  response  computation.  The 
initial  conditions  and  excitation  were  used  to  start  then  execute  a  random 
vibration  response  simulation  with  ANNs  in  the  space  of  u  (box  number  one 
on  the  second  line  of  Figure  4).  The  test  was  iterated,  i.e.,  the  estimated 
responses  at  step  j  were  used  as  initial  values  for  response  predictions  at  time 
indices  greater  tiian  j.  The  first  and  most  significant  column  of  u  from  the  test 


1622 


data  and  the  first  column  from  the  ANN  simulated  data  are  compared  in 
Figures  7a  and  7b.  This  is  the  dominant  component  of  the  response.  The  match 
is  good,  particularly  in  view  of  the  fact  that  the  simulation  is  iterated.  Note  that 
although  these  and  later  response  predictions  remain  fairly  well  in  phase  with 
the  test  responses,  this  is  not  usually  the  case.  Typically,  we  hope  that  the 
simulated  response  amplitudes  match  the  test  responses  well,  and  accept  the 
fact  that  phase  will  usually  be  lost.  Figures  8a  and  8b  compare  the  spectral 
densities  of  the  signals  shown  in  Figures  7a  and  7b.  (Tliese  were  comput^ 
using  the  technique  described,  for  example,  in  Wirsching,  Paez,  and  Ortiz, 
[14].)  A  block  size  of  256  data  points  was  used,  along  with  a  Hanning 
window,  and  an  overlap  factor  of  0.55.  The  first  harmonic  of  motion,  at  0.25 
Hz,  is  very  well  match^.  A  third  harmonic  of  motion  appears  to  be  present  in 
botii  the  test  and  simulated  signals;  the  BPN  provides  a  better  match  of  the 
third  harmonic  than  the  CNLS  net.  Figures  9a  and  9b  compare  the  kde’s  of  the 
signals  shown  in  Figures  7a  and  7b.  The  responses  are  clearly  non-Gaussian, 
as  anticipated  because  of  nonlinearity,  and  to  some  extent  the  simulated 
component  responses  match  the  character  of  the  test  response.  This  match 
needs  to  be  improved  to  maximize  the  quality  of  the  simulation.  (Some  of  the 
mismatch  is  caused  by  the  limited  data  -  1000  points  -  upon  which  the 
comparison  is  based.)  However,  as  will  be  seen,  the  simulated  synthesized 
responses  have  kde’s  that  match  test  response  kde’s  quite  well. 

Table  1.  Parameters  of  the  test  system. 


Tangent 

Maximum 

Viscous 

Index 

Mass 

Stiffness 

Deformation 

Damping 

1 

1.0 

40 

1.0 

0.40 

2 

0.95 

38 

1.0 

0.38 

3 

0.90 

36 

1.0 

0.36 

4 

0.85 

34 

1,0 

0.34 

5 

0.80 

32 

1.0 

0.32 

6 

0.75 

30 

1.0 

0.30 

7 

0.70 

28 

1.0 

0.28 

8 

0.65 

26 

1.0 

0.26 

9 

0.60 

24 

1.0 

0.24 

10 

0.55 

22 

1.0 

0.22 

11 

2 

1.0 

0.02 

The  simulated  responses  are  now  reconstructed  by  substituting  into  Eq.  (1) 
using  the  simulated  u  (box  number  four  on  the  second  line  of  Figure  4),  and 
the  results  are  compared  to  the  test  response  at  mass  10  (box  nuniber  five  on 
the  second  line  of  Figure  4).  The  result  from  the  BPN  simulation  is  shown  in 
Figure  10a;  the  result  from  the  CNLS  net  simulation  is  shown  in  Figure  10b. 
Of  course,  the  matches  are  quite  good  since  the  dominant  component  is  well 
simulated  by  both  ANNs.  Figures  11a  and  11b  compare  the  spectral  densities 
of  the  signals  shown  in  Figures  10a  and  10b.  The  spectral  densities  estimated 
from  the  simulated  signals  match  those  of  the  test  signals  well  up  to  the 
frequency  where  components  are  no  longer  simulated;  the  CNLS  net  does  a 
slightly  better  job  of  matching  the  test  spectral  density  than  the  BPN.  At  mass 
10  the  rms  response  of  the  test  signal  is  0.45  in,  and  the  rms  values  of  the 
simulated  responses  are  0.47  in  and  0.40  in  for  the  BPN  and  CNLS  net. 


1623 


respectively.  The  corresponding  kde’s  of  test  and  sinaulated  responses  were 
computed  and  are  shown  in  Figures  12a  and  12b.  The  probabilistic  character  of 
the  responses  appears  to  be  matched  well  by  the  ANN  simulations. 


q 


Figure  5.  A  nonlinear  spring-mass  system. 


Figures  6.  Response  at  mass  10  to  Gaussian  white  noise  input. 


Figures  7a  and  7b.  Comparison  of  test  and  ANN  simulated  responses  - 
Component  1.  BPN  simulation  on  left;  CNLS  simulation  on  right.  ANN 
simulation  -  solid  line;  test  data  -  dashed  line. 


1624 


10*’  10°  io’ 

Frequency,  Hz 


Figures  8a  and  8b.  Comparison  of  the  spectral  density  estimates  of  test  and 
ANN  simulated  responses  -  Component  1.  BPN  simulation  on  left;  CNLS  net 
simulation  on  right.  ANN  simulation  -  solid  line;  test  data  -  dashed  line. 


Figures  9a  and  9b.  Comparison  of  the  kernel  density  estimators  of  test  and 
ANN  simulated  responses  -  Component  1.  BPN  simulation  on  left;  CNLS  net 
simulation  on  right.  ANN  simulation  -  solid  line;  test  data  -  dashed  line. 


Figures  10a  and  10b.  Test  (dashed)  and  simulated  (solid)  responses  at  a  mass 
10  in  the  simple  system  -  BPN  (left),  CNLS  net  (right). 


1625 


10 


10 


Figures  11a  and  11b.  Test  (dashed)  and  simulated  (solid)  estimated  response 
spectral  densities  at  mass  10  in  the  simple  system  -•  BPN  (left),  CNLS  net 
(right). 


Figures  12a  and  12b.  Test  (dashed)  and  simulated  (solid)  response  kde’s  at 
mass  10  in  the  simple  system  -  BPN  (left),  CNLS  net  (right). 

Conclusions 

A  sequence  of  operations  leading  to  the  simulation  of  nonlinear  structural 
random  vibrations  with  ANNs  is  described  in  this  paper.  Such  simulations  are 
desirable  because  of  their  efficiency  and  relative  accuracy.  It  is  argued  that  if 
the  motions  can  be  decomposed  and  transformed  into  simple  components,  then 
the  simulation  will  be  simpler  and  more  accurate.  A  numeric^  example 
confirms  that  relatively  simple  motions  can,  indeed,  be  modeled  with  ANNs  - 
the  BPN  and  CNLS  net  (a  local  approximation  network)  in  particular.  Given 
sufficient  training  data  accurate  simulations  of  simple  components  should 
always  be  possible,  though  obtaining  satisfactory  accuracy  may  r^uire 
substantial  effort.  Accurate  simulations  should  correctly  reflect  probabilistic, 
spectral,  and  all  other  characteristics  of  the  simulated  component  responses. 
When  component  responses  are  correctly  modeled  then  system  level  responses 
will  be  simulated  accurately. 


1626 


References 


1.  Allemang,  R.,  Brown,  D.,  (1988),  “Experimental  Modal  Analysis,” 
Chapter  21  in  Shock  and  Vibration  Handbook,  Third  Edition,  Harris,  C., 
editor,  McGraw-Hill,  New  York, 

2.  Baldi,  P.,  Homik,  K.,  (1989),  “Neural  Networks  and  Principal 
Component  Analysis.  Learning  from  Examples  without  Local  Minima,” 
Neural  Networks,  2,  53-58, 

3.  Freeman,  J.,  Skapura,  D.,  (1991),  Neural  Networks,  Algorithms, 
Applications,  and  Programming  Techniques,  Addison-Wesley,  Reading, 
Massachusetts. 

4.  Golub,  G.  H.,  Van  Loan,  C.  F.,  (1983),  Matrix  Computations,  Johns 
Hopkins  University  Press,  Baltimore,  Maryland. 

5.  Haykin,  S.,  (1994),  Neural  Networks,  A  Comprehensive  Foundation, 
Prentice  Hall,  Upper  Saddle  River,  New  Jersey. 

6.  Homik,  K.,  Stinchcombe,  M.,  White,  H.,  (1989),  “Multilayer 

Feedforward  Networks  are  Universal  Approximators,”  Neural  Networks, 
V.  2,  359-366. 

7.  Hunter,  N,  (1992),  “Application  of  Nonlinear  Time  Series  Models  to 
Driven  Systems,”  Nonlinear  Modeling  and  Forecasting,  Casdagli,  M., 
Eubank,  S.,  eds.,  Santa  Fe  Institute,  Addison-Wesley. 

8.  Jones,  R.  D.,  et.  al.,  (1990),  "Nonlinear  Adaptive  Networks:  A  Little 
Theory,  A  Few  Applications,"  Cognitive  Modeling  in  System  Control,  The 
Santa  Fe  Institute. 

9.  Meirovitch,  L.,  (1971),  Analytical  Methods  in  Vibrations,  The  Macmillan 
Company,  New  York. 

10.  Moody,  J.,  Darken,  C.,  (1989),  “Fast  Learning  Networks  of  Locally- 
Tuned  Ifrocessing  Units,”  Neural  Computation,  V.  1,  281-294. 

11. Papoulis,  A.,  (1965),  Probability,  Random  Variables,  and  Stochastic 
Processes,  McGraw-Hill,  New  York. 

12.  Rosenblatt,  M.,  (1952),  "Remarks  on  a  Multivariate  transformation," 
Annals  of  Mathematical  Statistics,  23,  3,  pp.  470-472. 

13.  Silverman,  B.  W.(1986),  Density  Estimation  for  Statistics  and  Data 
Analysis,  Chapman  and  Hall. 

14. Wirsching,  P.,  Paez,  T„  Ortiz,  K.,  (1995),  Random  Vibrations  -  Theory 
and  Practice,  Wiley,  New  York. 

15.  Yamamoto,  K.,  (1992),  “Modeling  of  Hysteretic  Behavior  with  Neural 
Network  and  its  Application  to  Non-Linear  Dynamic  Response  Analysis,” 
Applic.  Artif  IntelL  in  Engr.,  Proc.  7th  Conf,  AING-92,  Comp.  Mech., 
UK,  pp.  475-486. 


1627 


1628 


Dynamic  Properties  of  Pseudoelastic  Shape  Memory  Alloys 

D.Z.  and  Z.C.  Feng*’ 

“Department  of  Engineering  Mechanics 
Hunan  University,  China 

^’Department  of  Mechanical  Engineering 
Massachusetts  Institute  of  Technology 
Cambridge,  Massachusetts  02139  USA 


ABSTRACT 

In  this  paper,  we  report  a  set  of  vibration  transmission  experiments  that  are  conducted  to  investigate  how  the 
pseudoelasticity  of  shape  memory  alloys  (SMAs)  affects  the  transmissibility  characteristics  of  a  spring-mass  system, 
where  a  shape  memory  alloy  rod  is  used  as  a  spring.  The  tests  are  conducted  by  subjecting  the  SMA  bar  under 
tension-compression  and  under  bending.  The  test  results  indicate  that  compared  with  ordinary  alloys,  SMAs  have  a 
much  higher  damping.  Most  importantly,  the  damping  property  depends  on  the  amplitude  of  the  responses  indicating 
that  the  spring-mass  system  is  nonlinear.  Furthermore,  the  high  damping  property  persists  to  the  high  frequency 
limit  (above  1  KHz)  permitted  by  the  equipment  setup. 

Keywords:  Shape  memory  alloys,  Nitinol,  vibration  damping,  hysteresis 

1.  INTRODUCTION 

Shape  memory  alloys  (SMAs)  such  as  Nickel-Titanium  (NiTi)  and  Copper-  Zinc-Aluminum  (CuZnAl)  exhibit  nonlin¬ 
ear  mechanical  properties.  Specifically,  in  a  temperature  environment  which  is  higher  than  the  phase  transformation 
temperature  of  the  material,  when  an  applied  stress  exceeds  a  certain  threshold,  stress  induced  phase  transformation 
generates  large  strains  in  the  material  so  that  the  material  can  accommodate  large  strain  with  little  change  in  the 
applied  load.  On  the  unloading  cycle,  a  reverse  phase  transformation  takes  place,  thus  no  permanent  deformation 
remains.  Moreover,  the  stress-strain  relationship  during  this  loading  and  unloading  cycle  shows  hysteresis.  This 
phenomenon  is  called  pseudoelasticity.  The  lower  elastic  modulus  and  higher  material  damping  of  SMAs  are  desir¬ 
able  characteristics  of  passive  vibration  control  systems.  Previous  investigations  show  that  the  damping  of  SMAs 
displays  a  peak  and  the  elastic  modulus  demonstrates  a  trough  in  the  vicinity  of  the  phase  transformation  during 
the  heating  and  cooling  processes  [l]-[3].  Some  researchers  have  pointed  out  that  the  pseudoelasticity  of  SMAs  can 
augment  passive  damping  significantly  in  structural  systems  and  SMAs  have  potential  application  in  passive  vibra¬ 
tion  control  [4]-[5j.  Recently,  a  research  study  has  been  undertaken  to  measure  acceleration  trainsmissibility  of  NiTi 
shape  memory  alloy  springs  (6). 

Among  all  of  these  studies  listed  above,  the  investigators  failed  to  realize  that  the  measurements  on  the  damping 
coefficient  and  Young’s  modulus  depend  on  the  specific  test  methods  used.  In  [7],  we  have  used  continuum  models 
of  pseudoelastic  SMA  model  to  illustrate  the  nonlinear  nature  of  the  dynamic  response  of  SMAs.  In  this  paper,  we 
present  experimental  evidence  on  the  nonlinear  behavior  of  the  materials. 

When  used  as  actuators,  SMAs  typically  have  a  very  slow  response  to  thermal  actuations  unless  SMAs  are  made 
into  very  fine  filaments.  For  this  reason,  SMAs  have  the  perception  of  being  limited  to  low  frequency  applications. 
But  when  used  as  a  passive  vibration  damper,  the  high  damping  property  is  not  limited  to  low  frequency  applications. 
Here,  we  demonstrate  the  high  damping  property  of  SMAs  to  a  high  frequency  permitted  by  our  equipment  setup. 

(Send  correspondence  to  Z.C.F.) 

Z.C.F.:  Email:  zfeng@mit.edu;  Telephone:  617-253-6345;  Fax:  617-258-5802 


1629 


Figure  1-  The  experimental  setup  of  the  tension-compression  test. 

2.  EXPERIMENTAL  SETUP  AND  PROCEDURE 

Our  vibration  tests  include  tension-compression  test  and  bending  test.  The  experimental  setup  for  these  two  types 
of  tests  are  similar.  Fig.  1  is  a  schematic  diagram  of  the  tension-compression  test  setup.  A  test  specimen  is  mounted 
vertically  by  a  clamping  fixture  attached  to  a  shaker.  Its  top  end  is  fixed  to  a  mass.  In  bending  test,  the  specimen  is 
fixed  horizontally  at  one  end  and  with  a  mass  of  ISOg  at  the  other  end.  The  base  harmonic  motion  is  generated  from 
HP3562A  Dynamic  Signal  Analyzer,  then  amplified  through  B&K  Power  Amplifier  2707  ajad  finally  provided  by  B&K 
Exciter  body  4801  with  general  purpose  head  4812.  Both  base  excitation  and  the  response  at  the  end  of  a  specimen 
are  picked  up  by  two  accelerometers  Picomin  22  and  PCB  309A  respectively.  The  signals  pass  through  two  Kistler 
5004  Charge  Amplifiers  to  be  feed  into  an  HP  3562A  Dynamic  Signal  Anal3^er  to  obtain  transmissibility  curves. 
The  transmissibility  curves  are  easily  acquired  from  the  frequency  response  measurement  by  a  sine  sweep  within  the 
frequency  range  of  interest.  During  each  measurement,  the  base  motion  level  is  kept  constant  and  recorded  in  voltage 
for  convenience.  Ail  test  specimens  are  circular  rods,  acting  as  springs  in  the  testing  mechanical  spring-mass  system. 
The  SMA  rods  are  binary  NiTi  alloys  obtmned  from  Shape  Memory  Applications  Inc.  of  Santa  Clara,  California. 
Steel  and  aluminum  rods  are  also  used  as  our  specimens  for  comparison.  For  each  specimen,  the  transmissibility 
properties  of  the  system  are  investigated  under  several  base  motion  levels  to  observe  the  nonlinearity  of  the  system 
dynamics.  All  tests  are  carried  out  at  room  temperature. 

3.  TENSION-COMPRESSION  VIBRATION  TRANSMISSIBILITY  EXPERIMENT 

The  parameters  of  specimens  are  listed  in  Table  1.  The  acceleration  transmissibility  curves  obtained  are  shown  in 
Fig.  2-Fig.  5,  where  the  mass  in  the  spring-mass  system  is  250g.  It  is  readily  seen  that  the  pseudoelasticity  of  SMAs 
does  have  significant  effects  on  vibration  transmissibility  characteristics.  The  NiTi  rods  behave  as  nonlinear  softening 
springs.  When  the  base  motion  level  increases,  the  resonant  peak  shifts  to  lower  frequency  and  the  peak  becomes 
lower.  This  indicates  that  the  equivalent  elastic  modulus  of  the  NiTi  rod  becomes  lower  and  its  material  damping 
becomes  higher  as  the  base  motion  level  (measured  by  the  input  voltage  to  the  shaker)  increases.  Higher  base  motion 
levels  result  in  larger  strains  in  the  rods.  For  example,  the  strain  amplitudes  in  the  NiTi  #2  rod  are  about  0.001%, 


1630 


SMAOI 


60; - 

’  0,3V 
50  h 

,  -  2.0V 

0  8.0V 


600  650  700  750 

Frequency  Hz 

Figure  2.  Tension-compression  vibration  transmissibility  curves  of  NiTi  Rod  #  1  at  different  base  motion  levels. 

0.008%,  0.015%  and  0.03%  corresponding  to  the  base  motion  levels  0.3V,  2.0V,  4.0V  and  8.0V  respectively.  This 
amplitude  dependence  of  the  elastic  and  damping  properties  of  the  material  is  significally  different  from  those  of 
conventional  alloys.  Fig.  4  and  Fig.  5  are  the  transmissibility  curves  of  a  steel  rod  and  an  aluminum  rod.  Under  two 
different  base  motion  levels,  the  transmissibility  remains  the  same.  Since  the  system  response  at  resonances  is  very 
big  owing  to  the  very  low  damping  of  the  materials,  our  experimental  setup  prevents  us  to  test  higher  input  levels. 
Quantitative  results  related  to  the  transmissibility  curves  are  listed  in  Table  2. 

The  nonlinearity  of  the  system  dynamics  is  apparent  in  the  amplitude  dependence  of  the  damping  ratio  and  the 
resonance  frequency.  This  is  seldom  appreciated  in  the  past.  Although  the  damping  ratio  and  the  material  Young’s 
modulus  are  often  measured  using  vibration  tests,  only  one  excitation  level  is  used  in  many  tests.  Thus  this  nonlinear 
phenomenon  is  often  not  noticed. 


Table  1.  Parameters  of  Test  Specimens  of 
Tension- Compression  Vibration 

Specimen 

Length  (mm) 

Diameter  (mm) 

Mass  (g) 

NiTi  #  1 

240 

5.86 

42.3 

NiTi  #  2 

250 

4.96 

31.6 

Steel 

617 

4.80 

86.9 

Aluminum 

716 

6.36 

61.2 

1631 


Figure  3.  Tension-compression  vibration  tfansmissibility  curves  of  NiTi  Rod  #  2  at  different  base  motion  levels. 


Frequency  Hz 

Figure  4.  Tension-compression  vibration  transmissibility  curves  of  steel  rod  at  different  base  motion  levels. 


1632 


60. 


Aluminum 


55r 

50- 

45 1- 
CD  I 
"40[- 

'3 

S  35- 

1 

2  30  - 

25- 

20 1- 
i 

isi- 


0.5V 

2.0V 


!  i 

I  \ 

/  1 

/  \ 


500  550 

Frequency  Hz 


Figure  5.  Tension-compression  vibration  transmissibility  curves  of  the  aluminum  rod  at  different  base  motion  levels. 


Table  2.  Some  Experimental  Results  of 
Tension-Compression  Vibration 

Specimen 

Base  Motion 
Level  (V) 

Resonant 
Frequency  (Hz) 

Modal  Damping 
Ratio  (%) 

Resonant  Peak 
Value  (dB) 

NiTi  #  1 

0.3 

684.2 

0.52 

39.6 

2.0 

673.0 

0.83 

35.5 

8.0 

650.0 

1.30 

30.9 

NiTi  #  2 

0.3 

613.0 

0.55 

38.8 

2.0 

596.5 

1.00 

32.1 

4.0 

58l0 

1.10 

29.2 

8.0 

551.9 

1.60 

25.6 

Steel 

0.5 

714.6 

0.09 

55.9 

2.0 

714.5 

0.10 

54.3 

Aluminum 

0.5 

533.3 

0.12 

55.6 

2.0 

533.2 

0.10 

5'5.9 

Our  results  listed  in  Table  2  show  that  SMAs  have  higher  dampings  at  a  frequency  above  500  Hz.  This  indicates 
that  the  pseudoelastic  properties  of  SMAs  persist  up  to  reasonably  high  frequencies  relevant  to  vibration  and  acoustic 
control.  It  is  well  known  that  the  higher  damping  of  the  materials  is  associated  to  the  movement  of  the  boundaries 
between  different  metallurgical  phases.  It  is  believed  that  such  movement  of  interfaces  takes  place  at  the  speed  of 
sound  in  the  solid.  Therefore,  the  higher  damping  properties  of  pseudoelastic  SMAs  are  expected  to  persist  up  to 
even  higher  frequencies. 

To  investigate  the  damping  properties  of  pseudoelastic  SMAs  at  even  higher  frequencies,  we  conducted  another 
set  of  tension-compression  tests.  To  acquire  higher  resonant  frequency  of  the  system,  we  shorten  the  specimens  and 
use  a  smaller  mass  of  150g.  The  lengths  of  the  specimens  of  NiTi  #  1,  NiTi  #  2  and  steel  rods  are  150mm,120mm  and 
230mm  respectively.  Fig.  6,  Fig.  7  and  Fig.  8  are  their  transmissibility  curves  measured.  It  is  evident  that  the  SMA 


1633 


40 


SMA  #1 


Figure  6.  Tension-compression  vibration  transmissibility  curves  of  NiTi  Rod  #  1  at  higher  resonant  frequency. 


rods  are  effective  as  vibration  isolators  up  to  a  frequency  over  IK  Hz.  The  limited  power  of  the  shaker  prevented  us 
from  experimenting  with  even  higher  frequencies.  Some  quantitative  measurements  related  to  the  transmissibility 
curves  are  given  in  Table  3. 


Table  3.  Some  Experimental  Results  of 
Tension-Compression  Vibration  at  Higher  Frequency 


Specimen 

Base  Motion 
Level  (V) 

Resonant 
Frequency  (Hz) 

Modal  Damping 
Ratio  (%) 

Resonant  Peak 
Value  (dB) 

NiTi  #  i 

0.3 

1028 

0.23 

39.6 

1.0 

1021 

0.51 

37.1 

2.0 

1011 

0.53 

34.8 

4.0 

997 

1.10 

32.4 

NiTi  #  2 

0.3 

1061 

0.39 

41.8 

1.0 

1052 

0.68 

37.7 

2.0 

1039 

0.83 

34.5 

4.0 

1016 

0.96 

31.0 

Steel 

0.3 

1504 

0.02 

70.2 

2.0 

1503 

0.01 

70.9 

Prom  Table  2  and  Table  3  we  can  see  that  the  SMA  specimens  have  damping  ratios  which  are  an  order  of  magnitude 
higher  than  those  of  steel  specimens.  When  the  SMA  specimens  are  subjected  to  maximum  base  excitation  delivered 
by  the  shaker  (which  has  100  pound  maximum  force),  the  peak  tension-compression  strain  remains  less  than  one 
percent.  Since  NiTi  materials  have  been  shown  to  have  very  good  fatigue  life  if  the  strain  is  less  than  2  percent  [8], 
it  is  thus  safe  to  say  that  even  higher  damping  ratio  can  be  achieved  in  practical  usage  if  the  SMA  component  is 
designed  to  operate  at  close  to  2%  peak  strain.  A  2%  peak  strain  seems  to  be  small,  but  it  is  nearly  an  order  of 
magnitude  higher  than  the  maximum  elastic  strain  in  many  conventional  metal  alloys. 


1634 


■  0.3  V 
4-  1.0  V 
40r  0  2.0  V 

X  4.0  V 


CO 

^35[- 


i  X 

g30l-  X 


Frequency  (Hz) 


Figure  7.  Tension-compression  vibration  transmissibility  curves  of  NiTi  Rod#  2  at  higher  resonant  frequency. 


Steel 


Frequency  (Hz) 


Figure  8.  Tension-compression  vibration  transmissibility  curves  of  the  steel  rod  at  higher  resonant  frequency. 


1635 


The  tension-compression  tests  at  large  strains  could  be  conducted  using  a  larger  shaker.  We  point  out,  however, 
that  in  conducting  such  tests,  lateral  (bending)  vibration  must  be  prevented.  Since  the  base  motion  is  also  a  source  of 
parametric  forcing  of  the  bending  modes,  the  tension-compression  tests  can  be  complicated  by  the  onset  of  parametric 
instability  of  the  bending  modes. 

4.  BENDING  VIBRATION  TRANSMISSIBILITY  EXPERIMENT 

To  examine  the  damping  properties  of  SMAs  under  larger  strains,  we  conducted  bending  tests.  In  the  bending 
tests,  the  resonance  frequency  (the  fundamental  mode)  is  much  lower  than  the  tension-compression  mode,  thus  the 
shaker  can  exert  a  much  larger  displacement.  The  parameters  of  specimens  are  listed  in  Table  4.  The  acceleration 
transmissibility  curves  are  shown  in  Fig.  9  -  Fig.  12.  The  same  qualitative  trend  as  the  tension-compression 
experiment  is  seen.  As  we  would  expect,  the  resonant  peaks  of  the  system  with  NiTi  rods  are  more  heavily  suppressed 
than  the  tension-compression  tests.  For  example,  the  resonant  peaks  of  the  systems  with  specimen  NiTi  #  1  and 
NiTi  #  2  drop  from  35.5  dB  to  18.9  dB  and  from  32.1  dB  to  14.4  dB  respectively  for  the  same  base  motion  level  of 
2  V.  Since  strains  generated  in  bending  tests  are  larger  than  in  tension-compression  tests  and  larger  strains  lead  to 
larger  areas  of  the  hysteresis  loop  in  the  stress-strain  curve  of  the  NiTi  rods,  higher  damping  and  lower  equivalent 
elastic  modulus  occur  in  the  NiTi  rod.  This  indicates  that  the  pseudoelasticity  of  SMAs  makes  SMAs  more  effective 
particularly  in  large-  amplitude  vibration  controls.  Unlike  the  tension-compression  vibration  tests,  the  acceleration 
transmissibility  curves  for  the  steel  and  the  aluminum  rods  at  different  levels  of  base  motion  no  longer  collapse  to  a 
single  one.  This  nonlinearity  is  not  caused  by  the  constitutive  laws  of  the  material  but  rather  by  the  geometric  and 
inertial  nonlinearity  owing  to  the  large  amplitude  of  the  vibration.  In  addition  the  compliance  of  the  fixture  may 
also  contribute  Co  the  nonlinearity.  Some  quantitative  measurements  related  to  the  transmissibility  curves  are  given 
in  Table  5. 


Table  4.  Parameters  of  Test  Specimens  of  Bending  Vibration 

Specimen 

Length  (mm) 

Diameter  (mm) 

Mass  (g) 

NiTi  #  1 

100 

5.86 

17.6 

NiTi  #  2 

80 

4.97 

10.1 

Steel 

200 

4.80 

28.2 

Aluminum 

r  150 

6.36 

12.8 

Table  5.  Some  Experimental  Results  of  Bending  Vibration 


Specimen 

Base  Motion 
Level  (V) 

Resonant 
Frequency  (Hz) 

Modal  Damping 
Ratio  {%) 

Resonant  Peak 
Value  (dB) 

NiTi  #  1 

0.1 

24.8 

2.2 

28.0 

0.3 

24.1 

2.9 

26.0 

1.0 

22.4 

4.4 

20.9 

2.0 

20.9 

4.6 

18.9 

NiTi  #  2 

0.1 

22.8 

3.5 

23.2 

0.3 

21.9 

4.5 

20.3 

1.0 

19.8 

6.8 

16.3 

2.0 

18.2 

7.1 

14.4 

0.1 

22.2 

0.3 

0.34 

43.8 

Aluminum 

0.1 

23.1 

0.097 

47.3 

0.3 

23.0 

0.17 

46.0 

Although  larger  peak  strains  can  be  achieved  in  the  bending  tests,  peak  strain  occurs  only  at  locations  farthest 
from  the  neutral  axis.  Perhaps  the  more  cost  effective  way  of  utilizing  SMAs  in  bending  vibration  control  is  to  place 
SMAs  away  from  the  neutral  axis  in  a  composite  beam. 


1636 


25 

Frequency  (Hz) 


Figure  10.  Bending  vibration  transmissibility  curves  of  NiTL  Rod  #  2  at  different  base  motion  levels 


1637 


15  20  25  30  35 

Frequency  (Hz) 

Figure  11.  Bending  vibration  transmissibility  curves  of  the  steel  rod  at  different  base  motion  levels 


ALUMINUM 

60 1 - ^ - 


I 

"inl- 


Frequency  (Hz) 

Figure  12.  Bending  vibration  transmissibility  curves  of  the  aluminum  rod  at  different  base  motion  levels 


5.  CONCLUSION 

Our  test  results  indicate  that  compared  with  conventional  alloys,  SMA  materials  have  a  much  higher  damping  ratio. 
In  fact,  the  material  properties  are  nonlinear.  We  observe  that  the  resonance  frequencies  and  the  damping  ratios 
are  dependent  on  the  excitation  levels.  When  conventional  parameters  such  as  the  damping  ratio  and  the  Young’s 
modulus  are  used  to  characterize  these  nonlinear  materials,  we  must  realize  that  these  parameters  will  be  dependent 
on  the  test  methods  and  procedures. 

As  we  would  expect  based  on  the  fact  that  the  stress- induced  phase  transformation  takes  place  at  the  speed 
of  the  sound,  the  higher  damping  associated  with  pseudoelasticity  persists  to  frequencies  high  enough  for  practical 
vibration  and  acoustic  applications. 


BIBLIOGRAPHY 

1.  S.  Ceresara,  A.  Giarda,  G.  Tiberi,  F.M.  Mazzolai,  B.  Coluzzi  and  A.  Biscarini,  Damping  characteristics  of 
Cu*Zn-Al  shape  memory  alloys.  Journal  de  Physique  IV,  Vol.l,  C4,  235-240,  1991 

2.  H.C.  Lin,  S.K.  Wu  and  M.T.  Yeh,  Damping  characteristics  of  TiNi  shape  memory  alloys.  Metallurgical  Transac¬ 
tions  A,  Vol.  24  A,  2189-2194,  1993 

3.  C.  Li  and  K.H.  Wu,  Systematic  study  of  the  damping  characteristics  of  shape  memory  alloys.  Smart  Material, 
SPIE  Proceedings,  Vol.  2189,  314-325,  1994 

4.  E.J.  Graesser  and  F.A.  Cozzarelli,  Shape  memory  alloys  as  new  materials  for  aseismic  isolation.  Journal  of 
Engineering  Mechanics,  Vol. 117,  No. 11,  2590-  2608,  1991 

5.  P.  Thomson,  G.J.  Balas  and  P.H.  Leo,  The  use  of  shape  memory  alloys  for  passive  structural  damping.  Smart 
Mater.  Struct.,  Vol.  4,  36-42,  1995 

6.  E.J.  Graesser,  Effect  of  intrinsic  damping  on  vibration  transmissibility  of  Nickel-Titanium  shape  memory  alloy 
springs.  Metallurgical  and  Materials  TVansactions  A,  Vol.  26A,  2791-2796,  1995 

7.  Z.  C.  Feng  and  D.Z.  Li,  Dynamics  of  a  mechanical  system  with  a  shape  memory  alloy  bar.  J.  Intell.  Mater.  Syst. 
and  Struct.  Vol.  7,  399-410,  1996, 

8.  C.  Liang  and  C.A.  Rogers,  Design  of  shape  memory  alloy  springs  with  applications  in  vibration  control.  J.  of 
Vibration  and  Acoustics.  Vol.  115,  129-  135,  1993. 


1639 


1640 


INVESTIGATION  OF  THE  REDUCTION  IN  THERMAL 
DEFLECTION  AND  RANDOM  RESPONSE  OF  COMPOSITE  PLATES 
AT  ELEVATED  TEMPERATURES  USING  SHAPE  MEMORY 

ALLOYS 


Z.  W.  Zhong  and  Chuh  Mei 
Department  of  Aerospace  Engineering 
Old  Dominion  University,  Norfolk,  VA  23529-0247 

ABSTRACT 

The  reduction  in  thermal  deflection  and  random  response  of  composite 
plates  with  embedded  shape  memory  alloy  fibers  (SMA)  at.  elevated 
temperatures  is  investigated.  The  stress-strain  relations  are  developed  for  a 
thin  composite  lamina  with  embedded  SMA  fibers.  Finite  element  equations 
and  computational  procedures  are  presented  for  composite  plates  with 
embedded  SMA  with  the  consideration  of  nonlinearities  in  geometry  and 
material  properties.  The  results  demonstrate  that  SMA  can  be  effective  in 
reduction  of  thermal  deflection  and  random  response  of  composite  plates  at 
elevated  temperatures. 


INTRODUCTION 

For  some  alloys,  a  given  plastic  strain  can  be  completely  recovered 
when  the  alloy  is  heated  above  the  characteristic  transformation  (austenite 
finish,  Tf )  temperature.  This  shape  memory  effect  phenomenon  is  attributed 
to  the  material  which  undergoes  a  change  in  crystal  structure  knovm  as  a 
reversible  austenite  to  martensite  phase  transformation.  The  solid-solid  phase 
transformation  also  gives  an  increase  in  Young’s  modulus  by  a  factor  of  three 
or  four  and  an  increase  in  yielding  strength  approximately  ten  times.  The 
transformation  temperature  can  be  altered  by  changing  the  composition  of  the 
alloy.  Many  alloys  are  known  to  exhibit  the  shape  memory  effect.  They 
include  the  copper  alloy  family  of  Cu-Zn,  Cu-Zn-X  (X=Si,  Sn,  Al,  Ga),  Cu- 
Al-Ni,  Cu-Au-Zn  and  the  alloys  of  Ag-Cd,  Au-Cd,  Ni-Al,  Fe-Pt  and  others 
[1].  Nickel-Titanium  alloys  (Ni-Ti  or  Nitinol)  are  the  most  common  SMA  [2]. 

SMA  has  been  used  as  actuators  for  active  control  of  buckling  of 
beams  [3]  and  shape  control  of  beams  [4],  It  is  also  being  investigated  in 
active  vibration  control  of  beams  [5],  rotating  disk/shaft  system  [6],  large 
space  structures  [7]  and  flow-induced  vibration  [8].  The  design  of  extended 
bandwidth  SMA  actuators  was  also  investigated  [9]. 

Composites  with  embedded  SMA  fibers  use  shape  memory  alloy  fibers 
as  reinforcements  which  can  be  stiffened  or  controlled  by  the  addition  of  heat. 


1641 


The  concept  of  active  damage  control  of  hybrid  material  systems  using  the 
SMA  as  embedded  induced  strain  actuators  has  been  proposed  [10].  Active 
vibration  control  of  flexible  linkage  mechanisms  using  SMA  fiber-reinforced 
composites  has  been  investigated  [11].  Acoustic  transmission  and  radiation 
control  by  use  of  the  SMA  hybrid  composites  was  presented  [12,  13]. 
Manufacturing  of  adaptive  graphite/epoxy  structures  with  embedded  Nitinol 
wires  was  recently  reported  [14].  Detailed  formulations  of  the  bending,  modal 
analysis  and  acoustic  transmission  of  SMA  reinforced  composite  plates  have 
been  presented  [15,  16]. 

A  limited  number  of  investigations  on  the  thermal  postbuckling 
deflection  of  SMA  fiber-reinforced  composite  plates  exists  in  the  literature.  A 
feasibility  study  on  reduction  of  thermal  buckling  and  postbuckling  deflection 
of  composite  plates  with  embedded  SMA  fibers  was  reported  [17]  and  the 
passive  control  of  random  vibrations  of  SMA  embedded  composite  plates 
using  the  analytical  continuum  method  was  presented  [18].  The  finite  element 
analysis  of  random  response  suppression  of  composite  panels  using  SMA  was 
recently  proposed  [19].  In  all  those  investigations,  the  material  nonlinearities 
were  not  considered. 

In  the  present  paper,  the  stress-strain  relations  for  a  thin  composite 
lamina  with  embedded  SMA  fibers  are  presented.  Governing  equations  for 
postbuckling  and  random  vibration  of  SMA  fiber-reinforced  composite  plates 
are  presented  using  the  finite  element  method.  Solution  procedures  using  the 
incremental  and  Newton-Raphson  iteration  methods  are  presented.  Numerical 
examples  are  given. 

CONSTITUTIVE  AND  FINITE  ELEMENT  FORMULATION 


Consider  a  thin  composite  lamina;  for  example,  graphite-epoxy, 
having  an  arbitrary  orientation  angle  and  with  SMA  fibers  embedded  in  the 
same  direction  as  the  graphite  fibers.  The  stress-strain  relations  for  such  a 
lamina  in  the  principal  material  directions  are  derived  in  Appendix.  For  a 
general  k-th  layer  with  an  orientation  angle  0,  the  stress-strain  relations,  Eqs. 
A17  and  A18,  become 


r  -  n 

/ 

a. 

1 

'  to 

1  * 

< 

k 

i 

e 

m 

y 

k 

m  ^ 

AT,T>T,  (1) 


and 


f 

a.* 

A 

Q* 

k 

< 

>  —  - 

< 

♦ 

►  AT 

V 

y^. 

k 

T<T. 


(2) 


1642 


where  [0*]  and  [Q]„are  the  transformed  reduced  stiffness  matrices  of  the 
composite  lamina  and  the  composite  matrix,  Ts  is  SMA  stimulating 
temperature  (austenite  start  temperature),  respectively.  The  resultant  force 
and  moment  vectors  of  the  SMA  fiber-reinforced  composite  plate  are  defined 
as 

(,{N},m)  =  Cjo},il,z)cb 


or 


PL 

'A* 

'T’W 

1^“'] 

^  ^  r 

iMj 

B* 

D^JIkJ  I 

iHl 

[Kj 

where  the  laminate  stiffness  [A*],  [B*]  and  [D*]  are  all  temperature 
dependent,  the  recovery  inplane  force  (A"*}  and  moment  {m'}  vectors  are 
dependent  on  temperature  and  prestrain  (see  Fig.  A2).  The  vectors  {N^}  and 
{M^}  are  the  force  and  moment  vectors  due  to  initial  stress  {oo}.  The 
recovery  and  the  thermal  inplane  forces  and  moments  due  to  the  SMA 
recovery  stress  and  the  temperature  change,  respectively,  are 


({a;},{m;})  =  J_  Na;  T>  T;  andnuH  T  <T^ 


({jV^}.Kr})  =  L“  [e  Ar(U>&.  T>T, 


(4) 

(5) 


=  e*  a;  AT{U)d^,  T<T, 


The  inplane  strain  and  curvature  vectors,  {s'^}  and  (k),  are  defined 
from  the  von  Karman  strain-displacement  relations  as 


W.x 

{win 

^  ^ 

; 

.+Lj/2 

>  +  < 

•  +  z\ 

7^' 

1 

to 

=  K}  +  {e;}  +  «}+4K}  (7) 


=  {s°}+z{k} 

where  wo  is  the  initial  deflection  (sum  of  the  converged  thermal  deflections 
from  previous  incremental  steps)  which  is  necessary  for  the  nonlinear  material 


1643 


properties  considered,  and  u  and  v  are  the  in-plane  displacements  and  w  is  the 
transverse  deflection  measured  from  the  initial  deflection  position. 

The  governing  equations  for  a  SMA  fiber-reinforced  composite  plate 
subjected  to  a  combined  thermal  and  random  excitation  loads  can  be  derived 
through  the  use  of  a  variational  principle.  The  system  equations  in  finite 
element  expression  can  be  written  in  the  form 


H  0 
0 


K,  K, 
Ks 


01  rr  oi  0 


0  0  0  0  0  0 


+ 


1 

0" 

1 ' 

_^mbo 

+ 

- 1 

i  o 

2 

0  J 

0 

0 

"^3 

0 

ojj 

p.(0l 

1 

1  IP.] 

\PL 

1 

1 

1  0  J 

L-*  mtiT 

r 

’|p' 

L  mr  ^ 

+ 

o 

io 

I 

(8a) 


or 

[M\m + m  -  [K^,r]+[K]+ [K.  ] + j + 

= {^(0) +{p^)-  {p:)-  {p.  } + {p^ro)  -  {p:o)  -  {p.o) 

(8b) 

where  [M]  is  the  system  mass  matrix,  [K]  and  (P)  are  the  system  stiffness 
matrices  and  the  load  vector,  respectively;  [K^]  and  are  geometric 

stiffness  matrices  due  to  the  recovery  stress  a*(  or  {A*})  and  thermal 
inplane  force  vector  respectively;  [Ka]  and  [Ko]  are  the  geometric 

stiffness  matrices  due  to  the  initial  stress  (cjo}  and  the  initial  deflection  {Wo}, 
respectively;  [Nl]  and  [N2]  are  the  first  and  second-order  nonlinear  stiffness 
matrices  which  depend  linearly  and  quadratically  upon  displacement  {W}, 
respectively;  [Nlo]  is  the  first-order  nonlinear  stiffness  matrix  which  is 
linearly  dependent  on  the  displacement  (W)  and  the  known  initial  deflection 
(Wo}.  The  subscripts  b  and  m  denote  bending  (including  rotations)  and 
membrane  components,  respectively;  subscripts  r,  c  and  o  represent  that  the 
corresponding  stiffness  matrix  or  load  vector  is  dependent  on  the  recovery 
stress  a*,  initial  stress  {oo}  and  initial  deflection  {Wo},  respectively;  and  the 
subscripts  B,  Nm  and  NB  indicate  that  the  corresponding  stiffness  matrix  is  due 
to  the  extension-bending  coupling  laminate  stiffness  [B],  membrane  force 
vectors  {Nm}  (=[A*]{s;,})  and  {Nb}  (=[B*]{k}),  respectively.  The  stiffness 
matrices  [K],  [Nl],  [Nlo]  and  [N2]  are  also  temperature  dependent  and  all  the 
matrices  in  Eq.  (8b)  are  symmetric.  Detailed  derivations  of  the  governing 
equations  and  expressions  for  the  element  matrices  and  load  vectors  are 
referred  to  [19-21]. 


1644 


SOLUTION  PROCEDURES 


Equation  (8b)  is  a  set  of  nonlinear  ordinary  differential  equation  with 
respect  to  time  t,  and  some  of  the  load  vectors  are  independent  to  time  t.  The 
solution  for  Eq,  (8b)  consists  of  a  time-dependent  solution  and  a  time- 
independent  solution  as 

{W}  =  {WJ  +  {W(r)},  (9) 

where  {Ws}  is  the  time-independent  solution  and  its  physical  meaning  is  the 
large  thermal  deflection,  and  {W(f)}j  is  the  time-dependent  dynamic  solution 
whose  physical  meaning  is  small  random  oscillations  about  the  static 
equilibrium  deformed  position  {Ws}. 

Thermal  Deflection  or  Postbuckling 

For  nonlinear  material  properties  of  SMA,  the  incremental  method 
should  be  employed.  This  implies  that  the  material  properties  are  treated  as 
constant  withhi  each  small  increment  of  temperature.  Substituting  Eq.  (9)  into 
the  system  equation  of  motion,  Eq.  (8b),  and  neglecting  the  higher  order  terms 
of  {W(r)}j  for  small  random  response,  two  sets  of  equations  can  be  obtained. 
One  is  the  time-independent  nonlinear  algebraic  equations  which  yield  the 
thermal  postbuckling  deflection  (Ws)  and  it  can  be  written  as 

=  {P.T)-{P:)-{Pa)  +  {P.To]-{P:)-{Pao} 

where  the  nonlinear  stiffness  matrices,  [Nl]s,  [Nljs  and  [N2]ss,  linearly  and 
quadratically  depend  on  the  thermal  displacement  {Ws}.  The  temperature 
dependent  nonlinear  material  properties  are  handled  with  small  temperature 
increments  of  AT,  and  the  material  properties  are  thus  considered  to  be 
constants  within  each  small  temperature  increment.  The  initial  deflection  and 
initial  stress  are  both  zero  at  the  first  temperature  increment.  The  effects  of 
initial  deflection  and  stress  are  thus  included  in  the  formulation.  One  effective 
approach  for  solving  Eq.  (10)  involves  the  application  of  Newton-Raphson 
iterative  method.  Thus,  the  i-th  iteration  Eq.  (10)  can  be  written  as 

=(AP),  (11) 

then  and  {AP};^^  are  updated  by  using 

={WJ,.  +  {AW},,,  (12) 

The  solution  process  seeks  to  reduce  the  imbalance  load  vector  {AP},  and 
consequently  {AW},  to  a  specified  small  quantity  (10"^  in  this  study).  The 
tangent  stiffness  matrix  and  the  imbalance  load  vector  are 

+  +  (13) 

and 


1645 


{API  =  {P^}  -  {Pr'}  -{PJ  +  [P^J-  iPrl)  -  {P^}  ,j  . 

-([/:]-[Ji:„^,]+[<]+[^j+[^j+MA'iL  +[Afi„L  +i[JV2]„,){r,}, 

where  the  nonlinear  stiffness  matrices  [Nljsi,  pSTloJsi  and  pSf2]ssi  are  evaluated 
using  The  total  thermal  deflection  is  the  sum  of  the  converged  {Ws} 

from  the  many  small  temperature  increments. 

Random  Response  Analysis 

The  other  equation  derived  from  Eqs.  (8b)  and  (9)  is  a  dynamic 
equation,  and  it  can  be  written  as 

mm)}, 

+  (m-[^«.r]  +  [<]  +  [^<,]  +  [^J  +  [^lL  +lN2l,W{,t)).  (15) 

=  {^(0} 

The  sum  of  stiffness  matrices  in  Eq.  (15)  is  exactly  the  converged  tangent 
stiffness  matrix  from  Eq.  (1 1).  Therefore,  there  is  no  need  to  assemble  the 
system  stiffness  matrices  by  considering  the  effects  of  SMA  recovery  stress, 
thermal  stress,  and  thermal  deflection  from  each  element  as  in  the 
conventional  finite  element  approach.  By  set  {P(t)}=0,  Eq.  (15)  is  a  standard 
linear  eigenvalue  problem,  and  the  natural  frequencies  and  mode  shapes  of 
vibration  about  the  thermally  deflected  position  are  obtained. 

Substituting  a  modal  transformation  and  truncation  of  [W],  =  [(|)]{g'} 
into  Eq.  (15)  with  the  consideration  of  damping,  a  set  of  uncoupled  modal 
equations  about  the  thermally  deflected  or  buckled  position  can  be  expressed 
as 

+  +  =  r=l,2,...,N  (16) 

where  the  modal  mass,  stiffness  and  force  are 

/.  =  {<!>,}’■  {7>(0}  {<!>,} 

Thus,  the  root  mean  square  (RMS)  of  maximum  deflections  and  strains  can  be 
easily  determined. 


RESULTS  AND  DISCUSSION 

The  results  shown  in  this  study  were  based  on  an  SMA  fiber- 
reinforced  composite  laminate,  where  the  graphite-epoxy  composite  was 
treated  as  matrix.  The  following  material  properties  were  used  in  the  analysis: 

SMA-Nitinol  Graphite-Epoxy 

T.  100°F(37.78°C)  E'  22.5(155) 

If  145(62.78)  Ez  1.17(8.07) 


1646 


Gi2  0.66(4.55) 


E*  From  Reference  [2] 
a  *  From  Reference  [2] 

G*  3.604  Msi  (24.9  GPa),  T<Ts 
G*  3.712  Msi  (25.6  GPa),  T>Ts 
0.3 

p  0.6067  xl0'"/6- 5^ /m\645QKg/m^) 

a  5.7  X 10-^  r  F(10.26  x  lO"^  E  Q 


|Xi2  0.22 

p  0.1458x10-' (1550.07) 

ai  -0.04xl0-®(-0.07xl0-®) 
Oil  16.7x10-^(30.1x10-") 


The  finite  element  used  in  this  investigation  is  the  three-node  Mindlin  (MIN3) 
plate  element  with  improved  transverse  shear  [22].  A  reference  or  ambient 
temperature  -  10''  F  and  a  uniform  temperature  distribution  are  used  in  all 
the  examples. 


Thermal  Deflection 

A  simply  supported  15x12x0.048  in.  rectangular  (0/90/90/0)s  laminate 
with  or  without  SMA  fibers  is  studied  in  detail.  The  mesh  size  is  10x8x2  for 
the  full  plate.  Figures  1  and  2  show  the  maximum  deflections  versus 
temperature  for  a  laminate  with  no  SMA  fibers  (vs=0)  and  with  10%,  20%, 
30%  SMA  fibers  and  3%  prestrain  (v^  =  10%,  20%,  30%  and  s,  =3%), 
respectively.  It  is  seen  that  the  thermal  deflection  of  the  panel  without  SMA 
approximately  reaches  2  times  the  plate  thickness  (Wmax/h=2)  at  200'’ F .  For 
the  panel  with  different  volume  fraction  of  SMA,  Fig.  2  shows  that  the 
thermal  deflection  drops  dramatically  after  the  SMA  is  activated  at 
transformation  temperature  (T>Ts)  and  the  deflection  will  increase  gradually 
when  the  thermal  expansion  effects  become  dominant.  The  most  important 
phenomenon  is  that  although  the  Young’s  modulus  of  SMA  is  lower  than  that 
of  the  composite  matrix  material,  the  thermal  deflections  of  the  panel  can  be 
reduced  because  of  the  effects  of  recovery  stress  of  SMA.  It  can  be  seen  that 
the  thermal  deflections  (Wmax/h)  are  less  than  1.0  at  200'’ jF  and  less  than  1.5 
at  300"  F .  Figure  3  shows  the  maximum  deflection  versus  temperature  for  a 
laminate  with  30%  SMA  fibers  and  3%,  4%  and  5%  prestrains.  Compared  to 
Fig.  2,  it  is  seen  that  the  volume  fraction  of  SMA  fibers  is  more  effective  than 
the  prestrain  of  SMA  in  the  reduction  of  thermal  deflection.  The  thermal 
deflections  versus  temperature  for  a  clamped  15x12x0.048  in.  rectangular 
(0/90/90/0)s  laminate  are  shown  in  Fig.  4.  The  clamped  plate  is  more  stiff  than 
the  simply  supported.  The  critical  buckling  temperature  for  the  clamped 
laminate  without  SMLA  is  slightly  higher  than  the  SMA  transformation 
temperature  Ts.  This  leads  to  the  postponement  of  thermal  deflection  until 
300"  F  for  the  clamped  laminate  with  10%  SMA  and  3%  prestrain.  In  this 
clamped  case,  this  implies  that  the  thermal  deflection  can  be  completely 
eliminated  for  the  highest  operating  temperature  less  than  300"F .  The 


1647 


thermal  deflection  in  the  temperature  range  between  the  critical  buckling 
temperature  and  SMA  activated  temperature  for  the  simply  supported  case 
shown  in  Figs.  2  and  3  can  also  be  suppressed  by:  1)  selecting  the  proper 
percentages  of  SMA  fiber  volume  fraction  and  prestrain,  and  2)  altering  the 
transformation  temperature Ts  by  changing  the  composition  of  alloy. 

Random  Response  .  n.  ,  a  u 

For  panels  with  SMA,  the  dynamic  response  is  affected  by  the 

components  of  stiffness  due  to  SMA  ([K])  and  due  to  recovery  stress  of  SMA 
(rX’*])  Note  that  the  Young’s  modulus  of  SMA  is  lower  than  that  of  the 
composite  matrix,  thus  the  panel  becomes  less  stiff  when  SMA  fibers  are 
embedded.  The  increase  in  the  dynamic  response,  observed  for  some  case,  is 
due  to  the  relatively  lower  modulus  and  higher  mass  density  of  the  SMA.  On 
the  other  hand,  large  inplane  tensile  forces  are  induced  by  the  recovery  stress 
of  SMA  and  this  effect  will  decrease  the  dynamic  response  of  the  paneL 
Figure  5  shows  that  the  RMS(Wm»A)  of  the  panel  with  10%  SMA  fibers  and 
5%  prestrain  at  170°F  is  slightly  larger  than  that  of  the  panel  without  SMA  at 
nO°F.  In  this  case,  the  recovery  forces  induced  ^e  not  sufficient  to 
overcome  the  loss  of  stiffness  due  to  the  modulus  deficiency  of  ShM  fibCTS. 
However,  the  panel  with  20%  and  30%  SMA  fibers  and  5%  prestrain  provide 
ample  recovery  forces  to  significantly  reduce  the  p^el  dynamic  response. 

Figure  6  shows  the  total  maximum  deflection  of  panel  with  no  SMA 
fibers  (vs=0)  and  for  three  SMA  prestrain  values  s,  =  3,  4  and  5%  for  each 

nonzero  SMA  volume  fraction  (vs=10,  20  and  30%)  at 
SSL=100dB.  It  clearly  indicates  that  the  six  graphite/epoxy  panels  with  SMA 

volume  fraction  of  either  20%  or  30%  and  s,  =  3,  4  and  5%  are  all 
acceptable  designs.  Compared  to  the  panel  with  no  SMA  fibers,  those  p^els 
give  much  small  amount  of  maximum  RMS  random  deflections  as  well  as 

thermal  deflections.  ,  .  r,  0+ mn 

The  power  spectral  density  (PSD)  of  the  maximum  deflection  at  100 

dB  is  shown  in  Fig.  7  for  three  cases;  no  SMA  at  10°  F ;  10%  SMA  and  3% 
prestrain  at  110°  F ;  and  30%  SMA  and  5%  prestrain  at  1 70°  F ,  respectively. 
It  is  seen  that  the  SMA  fiber-reinforced  plates  exhibit  significant  peak- 

amplitude  reduction  and  frequency  increase  at  170°  F . 

CONCLUDING  REMARKS 

The  stress-strain  relations  for  a  thin  composite  lamina  with  embedded 
SMA  fibers  have  been  developed.  The  finite  element  method  has  been 
successfully  implemented  to  analyze  the  thermal  deflection  and  random 


1648 


8 


response  of  SMA  fiber-reinforced  composite  plates  with  the  consideration  of 
nonlinear  material  properties  of  SMA  and  nonlinearity  in  geometry. 

With  the  proper  percentages  of  SMA  volume  fraction  and  prestrain 
and  also  the  altering  of  transformation  temperature  by  changing  the  alloy 
composition,  the  thermal  deflection  can  be  dramatically  reduced.  This 
reduction  in  thermal  deflection  could  be  useful  in  practical  applications  by 
maintaining  optimal  aerodynamic  configuration  for  flight  vehicles  and 
eliminating  snap-through  motions. 

The  RMS  maximum  deflection  can  be  reduced  with  some 
combinations  of  SMA  volume  fraction  and  prestrain.  After  the  SMA  is 
activated  and  the  recovery  forces  induced  are  sufficient  to  overcome  the  loss 
of  stiffness  due  to  the  modulus  deficiency  of  SMA,  the  dynamic  response  can 
be  significantly  reduced. 


ACKNOWLEDGMENTS 

The  authors  would  like  to  acknowledge  the  support  by  grant  F33615- 

91-C-3205,  AF  Wright  Laboratory.  Dr.  Howard  F.  Wolfe  is  the  technical 

monitor.  The  authors  would  also  like  to  thank  Dr.  Alex  Tessler,  NASA 

Langley  Research  Center  for  his  assistance  on  the  MIN3  element. 

REFERENCES 

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Review  of  Materials  Science,  Eds.  Huggins,  R.  A.,  Giordmaine,  J.  A. 
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2.  Cross,  W.  B.,  Kariotis,  A.  H.  and  Stimler,  F.  J.,  Nitinol  Characterization 
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3.  Baz,  A.  and  Tampe,  L.,  Active  Control  of  Buckling  of  Flexible 
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4.  Chaudhry,  Z.  and  Rogers,  C.  A.,  Bending  and  Shape  Control  of  Beams 
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5.  Baz,  A.,  Iman,  K.  and  McCoy,  L,  Active  Vibration  Control  of  Flexible 
Beams  Using  Shape  Memory  Actuators.  Journal  of  Sound  and  Vibration, 
1990,  Vol.  140,  pp.  437-456. 

6.  Segalman,  D.  J.,  Parker,  G.  G.  and  Inman,  D.  J.,  Vibration  Suppression  by 
Modulation  of  Elastic  Modulus  Using  Shape  Memory  Alloy.  Intelligent 
Structures,  Materials  and  Vibrations,  1993,  ASME-DE-Vol.  58,  pp.  1-5. 

7.  Maclean,  B.  J.,  Patterson,  G.  J.  and  Misra,  M.  S.,  Modeling  of  a  Shape 
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Structures.  Journal  of  Intelligent  Materials  Systems  and  Structures,  1991, 
Vol.  2,  pp.  72-94. 

8.  Kim,  J.  H.  and  Smith,  C.  R.,  Control  of  Flow-Induced  Vibrations  Using 
Shape  Memory  Alloy  Wires.  Adaptive  Structures  and  Material  Systems, 
1993,  ASME  AD-Vol.  35,  pp.  347-34. 

9.  Ditman,  J.,  Bergman,  L.  and  Tsao,  T.,  The  Design  of  Extended  Bandwidth 
Shape  Memory  Alloy  Actuators.  Proceedings,  AIAA/ASME  Adaptive 
Structures  Forum,  Hilton  Head,  SC,  1994,  pp.  210-220. 

10.  Rogers,  C.  A.,  Liang,  C.  and  Li,  S.,  Active  Damage  Control  of  Hybrid 
Material  Systems  Using  Induced  Strain  Actuator.  Proceedings,  32nd 
AIAA/ASME/ASCE/AHS/ASC  Structure,  Structural  Dynamics  and 
Materials  Conference,  Baltimore,  MD,  1991,  pp.  1190-1203. 

11.  Venkatesh,  A.,  Hilbom,  J.,  Bidaux,  J.  E.  and  Gotthardt,  R.,  Active 
Vibration  Control  of  Flexible  Linkage  Mechanisms  Using  Shape  Memory 
Alloy  Fiber-Reinforced  Composites.  Proceedings,  1st  European 
Conference  on  Smart  Structures  and  Materials,  Glasgow,  England,  Eds. 
Culshaw,  B.,  Gardiner,  P.  T.  and  McDonach,  A.,  Institute  of  Physics 
Publishing,  Bristol,  England,  1992,  pp.  185-188. 

12.  Anders,  W.  S.,  Rogers,  C.  A.  and  Fuller,  C.  R.,  Vibration  and  low 
Frequency  Acoustic  Analysis  of  Piecewise-Activated  Adaptive  Composite 
Panels.  Journal  of  Composite  Materials,  1992,  Vol.  26,  pp.  103-120. 

13.  Liang,  C.,  Rogers,  C.  A.  and  Fuller,  C.  R.,  Acoustic  Transmission  and 
Radiation  Analysis  of  Adaptive  Shape  Memory  Alloy  Reinforced 
Laminated  Plates.  Journal  of  Sound  and  Vibration,  1991,  Vol.  145,  pp. 
72-94. 

14.  White,  S.  R.,  Whitlock,  M.  E.,  Ditman,  J.  B.  and  Hebda,  D.  A., 
Manufacturing  of  Adaptive  Graphite/Epoxy  Structures  with  Embedded 
Nitinol  Wires.  Adaptive  Structures  and  Material  Systems,  1993,  ASME 
AD-Vol.  35,  pp.71-79. 

15.  Jia,  J.  and  Rogers,  C.  A.,  Formulation  of  a  Mechanical  Model  for 
Composites  with  Embedded  SMA  Actuators.  Proceedings,  18th  Biennial 
Conference  on  Failure  Prevention  and  Reliability,  Montreal,  Canada, 
1989,  ASME  DE-Vol.  16,  pp.  203-210. 

16.  Rogers,  C.  A.,  Liang,  C.  and  Jia,  J.,  Behavior  of  Shape  Memory  Alloy 
Reinforced  Composite  Plates,  Part  1;  Model  Formulation  and  Control 
Concept.  Proceedings,  30th  AIAA/ASME/ASCE/AHS/ASC  Structure, 
Structural  Dynamics  and  Materials  Conference,  Mobile,  AL,  1989,  pp. 
2011-2017. 

17.  Zhong,  Z.  W.,  Chen,  R.,  R.,  Mei,  C.  and  Pates,  C.  S.,  Ill,  Buckling  and 
Postbuckling  of  Shape  Memory  Alloy  Fiber-Reinforced  Composite  Plates. 
Symposium  on  Buckling  and  Postbuckling  of  Composite  Structures,  1994, 
ASME  AD-Vol.  41/PVP-Vol.  293,  pp.  115-132. 


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18.  Pates,  C.  S.,  IH.,  Zhong,  Z.  W.  and  Mei,  C.,  Passive  Control  of  Random 
Response  of  Shape  Memory  Alloy  Fiber-Reinforced  Composite  Plates. 
Proceedings,  Fifth  International  Conference  on  Recent  Advances  in 
Structural  Dynamics,  eds.  Ferguson,  N.  S.  et  al.  Institute  of  Sound 
Vibration  Research,  Southampton,  England,  July  18-21,  1994,  pp.  423- 
436. 

19.  Turner,  T.  L.,  Zhong,  Z.  W.  and  Mei,  C.,  Finite  Element  Analysis  of  the 
Random  Response  Suppression  of  Composite  panels  at  Elevated 
Temperatures  Using  Shape  Memory  Alloy  Fibers.  Proceedings,  35^^ 
AIAA/ASME/ASCE/AHS/ASC  Structure,  Structural  Dynamics  and 
Materials  Conference,  Hilton  Head,  SC,  1994,  pp.  136-146. 

20.  Xue,  D.  Y.  and  Mei,  C.,  A  Study  of  the  Application  of  Shape  Memory 
Alloy  in  Panel  Flutter  Control.  Proceedings,  Fifth  International 
Conference  on  Recent  Advances  in  Structural  Dynamics,  eds.  Ferguson,  N. 
S.  et  al.  Institute  of  Sound  Vibration  Research,  Southampton,  England, 
July  18-21,  1994,  pp.  412-422. 

21.  Chen,  R.  R.  and  Mei,  C.,  Thermo-Mechanical  Buckling  and  Postbuckling 
of  Composite  Plates  Using  the  MIN3  Elements.  Symposium  on  Buckling 
and  Postbuckling  of  Composite  Structures,  1994,  ASME  AD-Vol. 
41/PVP-Vol.  293,  pp.  39-53. 

22.  Tessler,  A.  and  Hughes,  T.  J.  R.,  A  Three-Node  Mindlin  Plate  Element 
with  Improved  Transverse  Shear.  Computer  Methods  in  Applied 
Mechanics  and  Engineering,  1985,  Vol.  50,  pp.  71-101. 

APPENDIX 

Stress-Strain  Relations  of  a  SMA  Embedded  Composite  Lamina 
A  representative  volume  element  of  a  SMA  fiber-reinforced  composite 
lamina  is  shown  in  Fig.  Al.  The  element  is  taken  to  be  in  the  plane  of  the 
plate.  The  composite  matrix,  for  example  graphite/epoxy,  has  the  principal 
material  directions  1  and  2  and  the  SMA  fiber  embedded  in  the  1 -direction. 

In  order  to  derive  the  constitutive  relation  for  the  1 -direction,  it  is 
assumed  that  a  stress  Oj  acts  alone  on  the  element  (Cj  =0)  and  that  the  SMA 
fiber  and  composite  matrix  are  strained  by  the  same  amount,  Sj  (i.e.,  plane 
sections  remain  plane),  the  1 -direction  stress-strain  relation  of  the  SMA  fiber 
can  be  described  as 

a,,  =  £>,+o;.  T>T,  (Al) 

or 

Qj,  =  £*(8i  -a.AT),  T  <f  (A2) 

where  Ts  is  the  austenite  start  temperature  and  as  is  the  thermal  expansion 
coefficient.  The  Young’s  Modulus  E]  and  the  recovery  stress  gI  are 


1651 


temperature  dependent,  indicated  by  superscript  (*).  The  recovery  stress  a  *  is 
also  dependent  on  the  prestrain  .  For  Nitinol,  a*  and  E*  can  be  determined 
from  Figs.  A2  and  A3,  respectively  [2].  Similarly,  the  one-dimensional  stress- 
strain  relation  in  the  1 -direction  for  the  composite  matrix  can  be  expressed  as 
=-E[»(Si -ottaAT)  (A3) 

The  resultant  force  in  the  1-direction  (02  =  0)  is  distributed  over  the 

SMA  fiber  and  composite  matrix  and  can  be  written  as 
a,.4,  (A4) 

where  (a  1, 4^1 ),  and  are  the  (stress,  cross  section  area)  of 

the  entire  element,  SMA  fiber,  and  composite  matrix,  respectively.  Thus,  the 
average  stress  a,  is 

<^1  (A5) 

where  v  =  A  /  A  and  =  A„  /  A,  are  the  volume  fractions  of 'SMA  and 
composite  matrix,  respectively.  When  T>Ts,  the  SMA  effect  is  activated  and 
the  one-dimensional  stress-strain  relation  in  the  1-direction  becomes 
a,  =  (£>,  +ct;)v^  -i- £i„(s,  -a,„Ar)v„ 

=  -Ei's,  +<J>.  AT” 


£:=£,„v„+£;v.  (A7) 

When  T<Ts,  the  SMA  effect  is  not  activated  and  the  stress  o,  is 
a,  =  £,‘6,  -  (E'a,v^  +  £,„a,„,v„,  )AT 

=  £,*(s,  -a, -AT) 
where 

A  similar  constitutive  relation  may  be  derived  for  the  2-direction  by 
assuming  that  the  applied  stress  02  acts  upon  both  the  fiber  and  the  matrix 
(a,  =  0) .  Thus,  the  one-dimensional  stress-strain  relations  in  the  2-direction 
for  the  SMA  fiber  and  the  composite  matrix  become 
C'2.  =  <^2  =  -E*  (82.  -  “.AT)  (A1 0) 

and 

'J2«=02=£2™(s2„-a,„Ar)  (All) 

respectively.  The  recovery  stress  does  not  appear  in  Eq.  (AlO),  since  the  SMA 
fiber  prestrain  s,.  and  recovery  stress  a  *  are  considered  to  be  a  1 -direction 
effect  only. 

The  total  elongation  is  due  to  strain  in  the  composite  matrix  and  the 
SMA  fiber  and  may  be  written  in  the  form 

As.  =  Ajs.^  +  As,,  (A12) 


1652 


Thus,  the  total  strain  becomes 
E2=S2„''„,+E2.'', 

Since  o,  =£*(£,, -a. AT^,  Eqs.  (AlO)  and  (All)  may  be  substituted  into 
Eq.  (A13)  to  give 


E' 


+  (a.v,+a,„v„)Ar 


Therefore,  the  modulus  and  thermal  expansion  coefficient  in  the  2-direction 
become 

£*  - -  (A15) 

and 

“2=a.2„i'„+a,v.  (A16) 

Expressions  for  the  hybrid  composite  Poisson’s  ratios  and  shear  moduli  follow 
from  similar  derivations. 

The  constitutive  relations  for  a  thin  composite  lamina  with  embedded 
SMA  fibers  can  be  derived  using  a  similar  engineering  approach  to  give 


\Q\\  Q\2 


0  Ql 


a, 

0 

0  ^ 

0 

v.  J 

y^AT,  T>T,  (A17) 


a; 

0  ' 

a;' 

\ 

Q22 

0 

< 

:S2 

>  _  < 

>AT 

0 

av 

v 

[Vl2, 

0  ^ 

/ 

=  O;  O;  0  s,  -  a,Ar,  T<T,  (A18) 

^  ^  See.V.Yuj  > 

where  [0]^  and  [Q*]  are  the  reduced  stiffness  matrices  of  the  composite 
matrix  and  the  composite  lamina,  respectively.  The  [Q*]  matrix  is  temperature 
dependent  and  is  evaluated  using  the  previously  derived  relations  as 

(■E*.  Ibz)  =  (£.».  i^.2..)‘'«  +  (C.  11.)'', 
and 

(E'  G')  =  - - .  (A20) 

where  the  |i’s  are  Poisson’s  ratios  and  the  G’s  are  the  shear  moduli.  The 
thermal  expansion  coefficients  a*  and  a2are  derived  in  Eqs.  (A9)  and  (A16). 


1653 


1654 


Vi%  0  10  20  30 

Figure  6,  Total  Maximum  Deflection/Thickness  at  HO^F  and  100  dB  SSL. 


Frequency  (Hz)  element  for  SMA  flber-relnforced 

Figure?.  Power  Spectral  Density  of  Wmax.  hybrid  composite  lamina. 


Temperature,  “F  Temperature, 


Figure  A2  -  SMA  recovery  stress  versus  temperaturepigure  A3  -  SMA  modulus  versus  temperature 
and  initial  strain  (Cross  et  al.,1970).  (Cross  et  al.,  1970). 


1655 


1656 


SIGNAL  PROCESSING! 


DESCRIPTION  OF  NON-LINEAR 
CONSERVATIVE  SDOF  SYSTEMS 


Michael  Feldman  Simon  Braun 

Faculty  of  Mechanical  Engineering 
Technion  —  Israel  Institute  of  Technology 
Haifa,  32000,  Israel 


1  INTRODUCTION 

There  are  several  methods  developed  for  the  analysis  and  identification  of  non-linear  dy¬ 
namic  systems  and  vibration  signals  [2,  8,  7,  9].  In  the  recent  past  the  characterization 
of  the  response  of  non-linear  vibration  systems  has  been  approached  using  the  Hilbert 
transform  in  time  domain  [4,  3].  The  objective  was  to  propose  a  methodology  to  identify 
and  classify  various  types  of  non-linearity  from  measured  response  data.  The  proposed 
methodology  concentrates  on  the  Hilbert  transform  signal  processing  techniques  essentially 
on  signal  envelope  and  instantaneous  frequency  extracting,  which  enables  us  to  estimate 
both  systems  instantaneous  dynamic  parameters  and  also  elastic  (restoring)  and  friction 
(damping)  force  characteristics.  When  the  traditional  phase  plane  (t/,y)  of  a  non-hnear 
system  is  replaced  by  the  complex  plane  (y,y),  the  signal  envelope,  the  instantaneous 
frequency,  and  the  obtained  backbone  takes  an  unusual  fast  osciEation  (modulation)  form 
[1].  In  addition,  the  obtained  envelope  of  the  signal  gains  a  bias  in  comparison  with  the 
phase  plane  ampHtude  [1].  This  naturaUy  occurring  fast  modulation  and  the  bias  requires 
more  detail  investigation.  Further  use  of  the  Hilbert  methodology  for  non-linear  structures 
also  invites  more  sophisticated  analysis. 

2  PHASE  PLANE  REPRESENTATION 

The  differential  equation  of  motion  for  SDOF  non-linear  conservative  vibration  system 
may  be  written  as  my  K{y)  =  0,  where  m  -  the  mass,  y  -  the  acceleration,  and  K{y) 
-  the  restoring  force  which  is  a  function  of  displacement  y.  The  restoring  force  contains 
the  linear  stiffness  and  also  any  additional  non-linear  restoring  force  component.  The 
second-order  differential  equation  of  a  conservative  system  will  then  take  the  general  form 

y  +  A:(i/)  =  0  (1) 


1657 


Figure  1:  Phase  plane  (dash  Hne)  and  Analytic  signal  (bold  line)  representation  of  Duffing 
equation  (e  =  5) 


where  the  term  k{y)  -  K{y)lm  represents  the  restoring  force  per  unit  mass  as  a  function 
of  the  displacement  y. 

Traditionally  we  introduce  a  new  variable  y,  which  enables  us  to  exclude  time  from 
the  equation  of  motion  although  y  and  y  are  still  time  dependent,  so  y  =  ^  =  ^y.  In  the 

new  coordinates  Eq.(l)  takes  the  Mowing  form:  f  =  ^-  Using  of  the  new  variable  y  is 
a  traditional  way  of  study  the  motion  of  an  oscillator  by  representing  this  motion  on  the 
y,  y  plane  (Fig.  1),  where  y  and  y  are  orthogonal  cartesian  coordinates  [2].  Hence,  the 
time  for  a  full  cycle  becomes 


where  ymax  -  the  maximum  value  of  the  displacement,  so  the  velocity  corresponding  to 
ymax  hi  an  extreme  position  is  zero,  /(*)  =  fc(*).  It  is  a  well  known  fact  that  the  expression 
obtained  for  T  is  a  function  of  y^ax,  thus  for  non-linear  systems  the  period  of  oscillation 
depends  on  the  total  stored  energy  (non-isochronism  of  oscillation  in  non-linear  system). 
We  shall  now  divide  the  angle  corresponding  to  the  quarter  of  a  complete  circle  into  n 
equal  small  sectors  dcf)  =  Next,  we  use  Eq.(2)  to  estimate  the  integral,  but  separately 
for  each  i  small  interval  with  new  limits;  from  //  =  co5[|  —  d(f>{i  —  1)]  to  —  cos(|  —  i) 


where  u  =  y  fym&x-  In-  this  case  only  for  a  linear  restoring  force  the  last  equation  results  in 
a  constant  value  27r.  In  general  case  of  non-  linear  system  the  current  period  is  a  varying 
function  of  a  phase  angle  {Ti{<l>)  i=^  const).  An  inverse  function  of  the  current  period  will  be 
called  an  curvtnt  angular  frequency  uJi{(l>)  =  The  current  angular  frequency  does  not 
remain  constant,  but  fluctuates  between  a  maximum  and  a  minimum  as  the  radius  vector 
rotates  between  the  y  -  and  y-axes  of  the  phase  plane.  Also  the  radius  vector  of  phase 
plane  r{4>)  becomes  to  modulate,  from  its  minimum  up  to  the  maximum  value  r{(j>)  = 


1658 


bm«.  [•! 

Figure  2:  Solution  (a)  and  periods  (b)  of  the  Duffing  equation  (e  =  5,  n  —  100,  bold  line 
-  the  current  period,  dash  line  -  the  mean  value  of  period 


+  y2(<^)  =  ^y(<^y -h  2 To  illustrate  this  interesting  phenomena  of  the 
current  period  modulation  let  us  consider  an  example  of  Duffing  equation. 

y {I  +  ey^)y  -  0  (^) 

where  e  is  the  non-linear  parameter.  For  the  purpose  of  calculating  the  current  pe¬ 
riod  of  free  vibration,  we  substitute  expression  Eq.(4)  into  Eq.(3)  and  obtain  T{i)  = 
4  Figure  2,  shows  the  free  vibration  signal  obtained  via  numeric 

V{i-«D(i+'(i+«D7^'  .  .  ,  ,  ,  w  j  j 

integration  of  the  Duffing  equation  together  with  the  calculated  current  period  ol  the  so¬ 
lution  for  the  total  number  of  small  sectors  n  =  100.  It  is  clear  that  the  the  current 
period  oscillates  two  times  faster  than  the  non-linear  solutions.  Now  it  is  clear  that  the 
known  direct  integration  for  the  total  period  Eq.(2)  produces  only  an  average  value  T 
which  corresporids  to  the  total  cycle  of  the  motion. 


3  COMPLEX  PLANE  REPRESENTATION  OF  AN¬ 
ALYTIC  SIGNAL 

Let  us  consider  another  important  technique  of  representation  of  vibration  process  in  the 
time  domain  on  the  base  of  rotating  vectors.  According  the  analytic  signal  theory  real 
vibration  process  t/(t),  measured  by,  say,  a  transducer,  is  only  one  of  possible  projections 
(the  real  part)  of  some  analytic  signal  F(£).  Then  the  second  projection  of  the  same 
signal  (the  imaginary  part)  y{i)  wiU  be  conjugated  according  to  the  Hilbert  transform. 
Using  the  traditional  representation  of  analytic  signal  y{t)  in  its  complex,  trigonometric 
or  exponential  form  Y{t)  =  y{t)  +  iy(t)  =  iy(0l  [  cos  m  +  j  sin  ^(t)  ]  =  A{t)  one 

can  determine  its  instantaneous  amplitude  ( envelope,  magnitude) 

A(t)  =  \Y{t)\  =  y/ym+m  =  (5) 

and  its  instantaneous  phase 

V,(f)  =  arctan  ^  =  Im[liiy(t)l  (6) 


1659 


The  instantaneous  angular  frequency  is  the  time- derivative  of  the  instantaneous  phase: 


u;{t)  =  -0(f)  = 


yWyji)  -  y(0y(0  _ 


Am 


=  Im 


Y{t) 


noj 


(7) 


Each  point  on  complex  plane  is  characterized  with  the  radius  vector  -A(f),  and  the  polar 
angle  0  (Fig.  1).  The  analytic  signal  method  is  an  expedient  effective  enough  to  solve  gen¬ 
eral  problems  of  vibration  theory,  among  them  -  analysis  of  free  and  forced  nonstationary 
vibrations  (transient  processes)  and  non-linear  vibrations. 


3.1  The  Hilbert  transform 

The  single-value  extraction  (demodulation)  of  an  envelope  and  other  instantaneous  func¬ 
tions  of  a  signal  is  an  issue  based  on  the  Hilbert  integral  transform  [7].  The  Hilbert  Trans¬ 
form  of  a  real- valued  function  a;(f)  extending  from  —  co  to  -j-oo  is  a  real-  valued  function 
defined  by  H[y(t)]  =  y(f)  =  J  dr,  where  3/(f)  -  the  Hilbert  transform  of  the  initial 

process  y{t),  and  meaning  of  the  integral  implies  its  Cauchy  principal  value.  Thus  y{t)  is 
the  convolution  integral  of  y{t)  with  (l/7rf),  written  as  y(f)  =  y{t)  *  (l/-7rf)  .  The  double 
Hilbert  transform  yields  the  original  function  with  an  opposite  sign,  that’s  it  carries  out 
shifting  of  the  initial  signal  in  — x.  The  power  (or  energy)  of  a  signal  and  its  Hilbert  trans¬ 
form  are  equal.  For  n[t)  lowpass  and  y{t)  highpass  signals  with  nonoverlapping  spectra 

[7], 

H[n(f)2/(f)]  =  7i(f)  y(f).  (8) 

3.2  Approximate  non-linear  system  representation 

The  showning  non-linear  restoring  force  k{y)  of  the  second-order  conservative  system  Eq.  1, 
can  be  recast  into  multiplication  of  varying  non-linear  natural  frequency  a;o(3/)  and  of  the 
system  solution  y+ojl{y)y  -  0.  Let’s  assume  that  the  varying  non-linear  natural  frequency 
uo{y)  could  be  separated  into  two  different  parts  [5].  The  first  part  o^o  is  much  slower  and 
the  second  component  ct>i(y)  is  faster  than  the  system  solution,  so  the  equation  of  motion 

will  be  _ 

y-\-[uj^+  ujl{y)  ]  y  =  0  (9) 

The  proof  of  the  decomposition  of  a  signal  into  a  sum  of  low-  and  high-pass  terms,  based 
on  Bedrosian’s  theorem  about  the  Hilbert  transform  of  a  product,  could  be  found  in  [6]. 
Now  according  to  nonoverlapping  property  of  the  Hilbert  transform  Eq.  (8  )  we  use  the 
Hilbert  transform  for  both  sides  of  Eq.  9  y  -f  u>Qy  -h  (^i{y)y  =  0.  Multiplying  each  side  of 
the  last  equation  by  j  and  adding  it  to  the  corresponded^ides  of  Eq.  9  we  get  a  differential 
equation  in  the  Analytic  signal  form  Y  4-  (jJqY  +  [  +  ywj  ]y  =  0,  where  Y  =  y~\-  jy.  This 

complex  equation  can  be  transformed  to  the  commonly  accepted  form 

y  +  j5oir  +  a;o\F  =  0  (10) 

where  Wqj  =  wJ  -h  -  the  varying  natural  frequency,  <^oi  =  -  the  fast 

varying  fictitious  friction  parameter,  and  Y  =  y  jy  -  is  the  complex  solution. 


1660 


The  obtained  equation  of  the  non-linear  system  has  the  varying  natural  frequency, 
consisting  of  slow  ZUq  and  fast  component,  and  also  has  the  fast  varying  fictitious  friction 
parameter  ^oi-  It  should  be  pointed  that  this  non-stationary  equation  is  not  a  real  equation 
of  motion.  It  is  just  an  artificial  fictitious  equation  which  produce  the  same  non-linear 
vibration  signal. 


3.3  Non-stationary  equation  of  motion  in  the  analytic  signal 
form 

Taking  into  account  the  analytic  signal  representations,  enables  one  to  consider  this  equiv¬ 
alent  equation  of  motion  so  as  to  estimate  instantaneous  natural  frequency  and  instanta¬ 
neous  damping  coefficient. 

Some  general  form  of  a  differential  equation  of  motion  in  the  analytic  signal  form  could 
be  written 

Y  -f-  2ho{A)Y  -f  u;l{A)Y  =  0  (U) 

where  Y  -  the  system  solution  in  the  analytic  signal  form,  ho  -  the  instantaneous  damping 
coefficient,  Wq  -  the  instantaneous  undamped  natural  frequency.  This  equation  of  motion 
will  have  varying  coefficients  that  satisfied  the  envelope  and  instantaneous  frequency  of 
non-linear  oscillated  solution. 

Solving  two  equations  for  real  and  imaginary  parts  of  Eq.(ll),  we  can  write  the  expression 
for  non-stationary  coefficients  as  functions  of  a  first  and  a  second  derivative  of  the  signal 
envelope  and  the  instantaneous  frequency. 


"oW=“  - 1  +  + a; 

,  ^  A  Cj 

Hi)  =  -7  - 


(12) 


where  ufo{t)  -  instantaneous  undamped  natural  frequency  of  the  system,  ho{t)  -  instan¬ 
taneous  damping  coefficient  of  the  system,  to,  A  -  instantaneous  frequency  and  envelope 
(amplitude)  of  the  vibration  with  their  first  and  second  derivatives  (cl;,  A,  A). 


3.4  Two-component  signal  representation 

Let  us  consider  a  conservative  system  having  a  single  degree  of  freedom  that  are  gov¬ 
erned  by  simple  non-linear  differential  equation  having  the  form  Eq.  1  Assuming  k  can  be 
expanded,  one  can  rewrite  Eq.  1  as  [8] 

n-l 

where  an  =  and  denotes  the  nth  derivative  with  respect  to  the  argument. 

We  assume  that  the  solution  of  Eq.  13  can  be  represented  by  an  expansion  having  the 
form  y{t,  e)  =  £yi(t)  +  £^y2(t}  +  £%(t)  +  - .  •  where  the  y^  are  independent  of  e.  Retaining 


1661 


Table  1:  Formulas  for  extreme  and  mean  values  of  instantaneous  frequency  and  amplitude 
of  bibarmonics 


Inst,  amplitude 

Inst,  frequency 

•^min  “  -^1  •^2 

Am&x  =  Al  A  A2 

A  =  Ai^l  +  Al/Al 

two  first  main  terms  of  the  solution  we  will  get  an  equation  where  each  of  amplitude  and 
frequency  is  a  function  of  linear  and  non-linear  parameters  of  the  system  (Eq.  13). 

y(t'^  =  +  2/2  =  -^1  cos  Wit  -f  A2  cos  W2t  (14) 


Universally  known  that  the  more  close  the  frequency  of  the  first  term  to  the  precise  solution 
27z JT  (Eq.  2)  the  better  approximation. 

Consider  the  non-linear  solution  that  consists  of  two  quasi-harmonics  each  one  with 
different  amplitude  and  frequency  in  time  domain.  In  this  case,  the  signal  can  be  modeled 
as  a  weighted  sum  of  two  monocomponent  signals,  each  one  with  its  own  instantaneous 
frequency  and  amplitude  function.  Assuming  that  each  individual  signal  has  a  large  band¬ 
width  and  time  duration  product  (with  a  purely  positive  IF),  application  of  the  Hilbert 
transform  for  both  sides  of  Eq.(14)  will  produce  an  analytic  signal  of  the  form. 

r(t)  =  +  (15) 


The  envelope  and  the  instantaneous  frequency  of  double- component  vibration  signal 
Y{t)  are: 


A{t)  =  {Al  +Al-\-  2A1A2  cos[(a;2  -  a;^)^]}^/^ 


a;(t)  =  ctJi  -i- 


Al  -i-  A1A2  cos[(a;2  -  ^i)t] 
(w2-c,i)-^A^it) 


(16) 

(17) 


From  Eq.(l6)  it  can  be  seen  that  signal  envelope  consists  of  two  diiferent  parts,  that 
is  a  constant  part  included  sum  of  component  amplitudes  squared  Al,  Al  and  also  a 
fast  varying  (oscillating)  part,  the  multiplication  of  these  amplitudes  with  function  cos  of 
relative  phase  angle  between  two  components.  The  same  inference  could  be  made  about 
the  instantaneous  frequency  of  the  two-terms  solution.  The  obtained  formulas  for  extreme 
and  mean  values  of  instantaneous  frequency  and  amplitude  of  the  biharmonics  are  shown 


in  Table  1. 

It  is  significant  that  the  mean  value  of  the  instantaneous  amplitude  is  very  close  to  the 
corresponded  amplitude  of  the  first  harmonics  (A  Ai).  Also  mean  value  of  the  instanta¬ 
neous  frequency  agrees  closely  with  the  frequency  of  the  first  harmonics  {w  ^  wi).  Again 
consider  an  example  of  the  Duffing  equation  (Eq.  4).  Using  for  instance  the  method  of  the 
analytic  signal  representation  [9]  one  can  get  an  equation  for  the  first  order  approximation 
of  the  free  vibration  frequency 

a.?  =  1  +  ^  (18) 


1662 


Angulw  fr*qu«ncy 

Figure  3:  Backbone  of  the  Duffing  system  (s  =  5) 


b 


Figure  4:  The  solution  (a)  and  the  instantaneous  frequency  (b)  of  the  Duffing  equation 
(e  =  5) 


where  a  —  j/max  is  some  maximum  of  the  displacement.  This  approximation  of  free  vi¬ 
bration  frequency  (dot  line  on  Fig.  3)  maps  an  approximate  backbone  very  close  to  the 
traditional  precise  backbone  (bold  line  on  Fig.  3). 

Let  now  an  approximate  solution  be  in  a  form  which  includes  the  first  two  harmonics  only 

y[t)  =  yi{t)  +  y2{t)  -  Aicosujit  -f  A2  cos  aj2t  (19) 

where  0^2  =  3a;i,  A2  =  Ai  -  I  -  A2  ,  K  -  numerical  coefficient.  The  instantaneous 
frequency  and  the  envelope  of  the  solution  derived  from  the  Hilbert  transform  takes  the 
form  of  oscillated  functions  with  double  frequency  (Fig.  4).  The  approximate  double 
component  solution  {e  =  5,  K  =  156)  coincides  very  closely  with  the  numerical  solution 
(Fig.  4,  a).  The  corresponded  envelope  (dash-dot  line),  and  instantaneous  frequency  (b) 
of  the  Duffing  equation  is  also  plotted  in  Figure  4. 

Maximum  values  of  the  envelope  A{t)  (Eq.  16)  are  equal  to  the  chosen  maximum  of 
displacement  a,  but  all  other  points  of  the  envelope  (dash-dot  hne  on  Fig.  4,  a)  including 
the  average  value  of  amplitude  A  are  smaller  than  the  maximum  of  displacement  a.  Now 
it  is  clear  that  for  non-hnear  conservative  vibrations  two  different  estimations  of  amplitude 
could  be  suggested.  The  first  estimation  is  the  maximum  of  displacement  which  is 


1663 


equal  to  the  maxiiiium  of  envelope  y^aax  ~  ^max-  The  second  estimation  is  the  average  value 
of  envelope  A.  The  difference  between  the  average  value  of  envelope  A  and  the  maximum 
of  displacement  a  could  be  considered  as  a  small  bias  of  amplitude  of  the  solution. 

It  is  vital  to  note  that  the  mean  value  of  the  instantaneous  frequency  Q  obtained 
through  the  Hilbert  transform  is  equal  to  the  theoretical  value  of  the  free  vibration  fre¬ 
quency  (Eq.  2  ).  On  Fig.  4,  b  the  dash  line  shows  these  two  coincident  functions.  Q  =  2Tr/T 
This  fact  indicates  that  the  Hilbert  transform  is  best  representation  of  non-linear  vibra¬ 
tions, 

3.5  Free  vibration  frequency  and  amplitude  dependence 

Because  we  considered  several  different  representations  of  vibration  amplitude  it  is  evident 
that  we  get  some  different  backbone  depictions.  A  traditional  theoretical  precise  backbone 
of  non-linear  system  is  a  dependency  between  the  average  free  vibration  frequency  a>  (cor¬ 
responded  to  the  average  total  cycle  of  vibration,  Eq.  2)  and  the  maximum  displacement 
Vmax  (Fig.  3,  bold  line).  For  a  conservative  system  with  a  chosen  maximum  displacement, 
the  theoretical  backbone  is  no  more  than  a  point  (-}-  on  Fig.  3).  As  far  as  we  can  use  the 
mean  value  of  the  envelope  A,  a  dependency  between  this  mean  value  and  the  average 
free  vibration  frequency  cD  could  be  a  new  kind  of  the  theoretical  backbone  (dotted  line  on 
Fig.  3).  But  the  difference  between  these  two  backbone  definitions  is  very  small  and  cor¬ 
responds  to  the  small  bias.  According  to  the  traditional  vibration  system  representation, 
conservative  non-linear  system  has  a  constant  amplitude.  It  means  that  vibrations  of  a 
conservative  Duffing  system  should  map  on  backbone  plot  just  as  a  point  with  a  constant 
amplitude  and  frequency. 

A  somewhat  different  backbone  for  a  conservative  system  is  gained  according  to  the  in¬ 
stantaneous  frequency  and  amplitude  representation.  Eliminating  the  time  dependent  os¬ 
cillating  part  cos[{uj2-^i)t]  from  Eq.(16)  and  Eq.(17),  we  shall  find  the  equation  between 
signal  instantaneous  characteristics  (envelope  and  frequency)  and  initial  four  parameters 
of  signal  components 

(20) 

uj\  —  2c4; 

Equation  (20)  determines  the  signal  envelope  A  as  a  function  of  instantaneous  frequency 
ui  in  the  form  of  an  hyperbola,  whose  length  and  curvature  depends  on  four  initial  pa¬ 
rameters  of  the  biharmonics.  So  according  to  the  Hilbert  transform  the  instantaneous 
backbone  of  a  non-linear  conservative  system  stretches  to  a  very  short  hyperbola  (dashline 
on  Fig.  3).  This  short  length  of  hyperbola  is  the  same  for  both  the  numerical  solution  and 
the  approximate  double  component  solution.  The  hyperbola  goes  very  close  both  to  the 
traditional  theoretical  point  of  maximum  displacement  (+)  and  to  the  approximate  mean 
value  of  envelope  (Fig.  3,  o).  Figure  3  also  includes  two  additional  backbones  (short  dash 
lines)  corresponding  to  different  maximum  values  (a)  of  the  displacement. 

3.6  Real  non-linear  force  estimation 

The  proposed  direct  time  domain  method  based  on  the  Hilbert  transform  allows  a  direct 
extraction  of  the  linear  and  non-linear  parameters  of  the  system  y  -f  uJoivYy  =  0  Horn  the 


1664 


Displacvmvnt 

Figure  5:  Cubic  force  characteristics  of  Duifing  model  (e  =  5) 

measured  time  signal  y  of  output.  The  resulting  non-linear  algebraic  equations  (Eq.l2) 
are  rather  simple  and  do  not  depend  on  the  type  of  non-hnearity  that  exists  in  the  struc¬ 
ture.  When  applying  this  direct  method  for  transient  vibration,  the  instantaneous  modal 
parameters  are  estimated  directly,  with  the  corresponding  equation 

y  +  ^oiV  +  ^oiV  ~  ^ 

where  ioh  -  is  the  fast  varying  natural  frequency,  5oi  “  is  the  fast  varying  fictitious  friction 
parameter,  y  -  is  the  displacement,  and  y  -  is  the  Hilbert  transform  of  the  displacement. 
With  this  representation  of  a  non-linear  solution  we  can  try  to  solve  an  inverse  identifi¬ 
cation  problem,  namely,  the  problem  of  estimation  of  the  initial  non-linear  elastic  force 
characteristics. 

3.6.1  Decomposition  technique 

Let  us  consider  the  case  of  a  conservative  system  with  an  initial  non-linear  spring.  Accord¬ 
ing  to  Eq.  11  this  real  non-linear  elastic  force  wiU  produce  two  different  fictitious  members 
(elastic  and  hysteretic  damping).  The  new  restoring  force,  here  assumed  to  be  a  function 
of  displacement  y  and  its  Hilbert  transform  y.  This  restoring  force  includes  both  the  fast 
hysteretic  damping  6oiy  and  the  fast  elastic  force  u^l-^^y.  It  means,  that  the  initial  non¬ 
linear  spring  force  <jJo{yyy  is  split  into  two  terms,  and,  by  summing  over  the  terms,  the 
initial  non-linear  force  characteristics  can  be  extended.  Therefore  a  simple  composition  of 
these  two  members  of  equation  of  motion  (Eq.  21)  will  result  in  the  real  non-linear  force 

characteristics: 

k[y{t)]  =  2ho{t)y-^L,l{t)y  (22) 

where  k[(y)]  is  the  real  elastic  instantaneous  force,  /io(0  instantaneous  damping 

coefficient,  Wo(0  is  the  instantaneous  undamped  natural  frequency.  Fig.  5  shows  an  exam¬ 
ple  of  the  instantaneous  elastic  force  identification  for  the  Duffing  system  (Eq.  4).  Fig.  5 
includes  the  results  of  the  identification  according  to  formula  Eq.  22  together  with  the 
initial  cubic  force  characteristics  k{y)  =  (1  +  5y^)y,  but  these  lines  agree  so  closely  that 
^Kere  is  no  a  difference. 


1665 


3.6.2  Scaling  technique 

Instead  of  the  previous  identification  of  the  detail  force  characteristics,  we  can  express  only 
the  relation  between  the  maximum  of  elastic  force  and  the  maximum  of  displacement.  It 
can  be  determined  by  following  consideration.  The  total  energy  of  a  conservative  vibration 
system  is  constant.  During  free  vibration  of  the  corresponded  fictitious  model  (Eq.  11),  for 
each  moment  the  energy  is  partly  kinetic,  partly  potential,  and  partly  fictitious  alternating 
positive  or  negative  damping. 

To  estimate  maximum  of  elastic  force  we  can  find  time  points  when  all  energy  is  stored 
in  the  form  of  strain  energy  in  elastic  deformation  and  the  fictitious  damping  energy  is 
zero.  Using  for  instance  the  biharmonics  representation  of  non-linear  vibrations  (Eq.l5) 
one  can  show  that  these  time  points  correspond  to  the  maximum  of  displacement.  This 
is  an  important  conservative  vibration  system  property,  that  around  every  peak  point  of 
the  displacement  the  corresponding  value  of  the  velocity  is  equal  to  zero  and  vice  versa. 
Therefore  around  every  peak  point  of  the  displacement  the  contribution  of  the  velocity  in 
the  varying  instantaneous  elastic  force  is  negligibly  small:  y{ti)  =  A(ti),  y{ti)  =  0.  The 
number  of  these  peak  points  is  far  less  than  a  total  number  of  points  of  a  vibration  signal. 
Therefore  it  could  be  recommended  to  extract  the  envelope  of  the  fictitious  elastic  force 
ufQ{t)y  to  obtain  the  average  value  of  the  envelope.  The  obtained  average  envelope  has 
a  small  bias  relative  to  maximums  of  the  spring  force.  Using  the  Hilbert  transform  and 
the  obtained  expressions  we  can  extract  the  maximum  values  of  non-linear  elastic  force 
corresponded  to  the  maximum  of  displacement  (circle  and  star  point  on  Fig.  5). 


4  CONCLUSIONS 

We  can  draw  the  following  conclusions  from  the  analytic  signal  representation. 

Whatever  the  method  of  non-linear  vibration  representation,  both  the  instantaneous  fre¬ 
quency  and  the  amplitude  of  free  vibration  is  a  complicated  signal.  Non-linear  solution 
could  be  represented  by  an  expansion  of  members  with  different  frequencies  or  by  a  time 
varying  signal  with  oscillated  instantaneous  frequency  and  envelope. 

The  instantaneous  frequency  and  envelope  of  non-linear  vibration  obtained  via  the  Hilbert 
transform  are  time  varying  fast  oscillating  functions.  For  example  in  the  presence  of  a  cubic 
non-linearity  and  a  threefold  high  harmonics,  the  frequency  of  the  instantaneous  parameter 
oscillation  is  twice  that  the  main  frequency  of  vibration. 

The  mean  value  of  the  instantaneous  frequency  obtained  through  the  Hilbert  transform 
is  equal  to  the  theoretical  total  cycle  average  value  of  the  free  vibration  frequency.  The 
maximum  value  of  vibration  envelope  is  equal  to  the  maximum  of  displacement.  These 
facts  indicates  that  the  Hilbert  transform  is  one  of  the  best  representations  of  non-linear 
vibrations.  Naturally  there  is  a  small  difference  (the  bias)  between  the  average  value  of 
the  envelope  and  its  maximum  value. 

The  dependency  between  the  average  envelope  and  the  average  instantaneous  frequency 
plots  the  backbone  that  practically  coincides  with  the  theoretical  backbone  of  non-linear 
vibrations. 

Using  the  proposed  Hilbert  transform  analysis  in  time  domain  we  can  extract  both  the 
instantaneous  undamped  frequency  and  also  the  real  non-linear  elastic  force  characteristics. 


1666 


References 


[1]  P.  Adamopoulos,  W.  Fong  and  J.K.  Hammond.  Envelope  and  instantaneous  phase 
characterisation  of  nonlinear  system  response.  Proc.  of  the  VI  Int.  Modal  Analysis 
Conf.,  1988,  pp.  1365  -  1371 

[2]  A.A.  Andronov,  A. A.  Vitt,  and  S.E.  Khaikin.  Theory  of  oscillators.  Pergamon  Press, 
1966,  815  p. 

[3]  M.  Feldman.  Non-linear  system  vibration  analysis  using  Hilbert  Transform  —  I:  Free 
vibration  analysis  method  ‘FREEVIB’,  Mechanical  Systems  and  Signal  Processing, 
1994,  8(2),  pp.  119  -  127. 

[4]  M.  Feldman,  S.  Braun.  Analysis  of  typical  non-Hnear  vibration  systems  by  using  the 
Hilbert  transform.  Proc.  of  the  XI  Int.  Modal  Analysis  Conf.,  Kissimmee,  Florida, 
1993,  pp.  799  -  805. 

[5]  M.  Feldman,  S.  Braun.  Non-linear  spring  and  damping  forces  estimation  during  free 
vibration.  Proc.  of  the  ASME  Fifteenth  Biennial  Conference  on  Mechanical  Vibration 
and  Noise  Conf.,  Boston,  Massachusetts,  1995,  V  3,  1241-1248  pp. 

[6]  Stefan  L.  Hahn.  The  Hilbert  Transform  of  the  Product  a{t)cos{ujot  -j-  4>o)-  Bulletin  of 
the  Polish  Academy  of  Sciences,  Technical  Sciences,  Vol.  44,  No  1,  1996,  pp. 75-80. 

[7]  Sanjit  K.  Mitra  and  James  F.  Kaiser.  Handbook  for  digital  signal  processing.  Wiley- 
Interscience,  1993,  1268  p. 

[8]  AH  H.  Nayfeh.  NonHnear  oscillations.  Wiley-Interscience,  1979,  704  p. 

[9]  L.  Vainshtein,  D.  Vakman.  Frequencies  separation  in  the  theory  of  vibration  and  waves 
(in  Russian).  1983,  Moscow,  Nauka,  228  p. 


1667 


1668 


A  RATIONAL  POLYNOMIAL  TECHNIQUE  FOR 
CALCULATING  HILBERT  TRANSFORMS 


N.E.King 

Ford  Motor  Company  Ltd. 

Research  and  Engineering  Centre 

Laindon 

Basildon 

Essex  SS15  6EE 

United  Kingdom 


K.Worden 

Dynamics  Research  Group 

Department  of  Mechanical  Engineering 

University  of  Sheffield 

Mappin  Street 

Sheffield  SI  3JD 

United  Kingdom 


Abstract 

This  paper  presents  a  new  technique  for  calculating  Hilbert  trans¬ 
forms  of  nonlinear  system  Frequency  Response  Functions  (FRFs). 

The  method  employs  rational  function  and  pole-zero  decomposi¬ 
tions  of  the  FRF.  The  new  method  allows  Hilbert  transforms  for 
zoomed,  or  generally  truncated,  data  without  the  use  of  correction 
terms.  The  method  is  validated  using  a  computer  simulation  of  a 
Single  Degree- Of- Freedom  (SDOF)  nonlinear  oscillator. 

INTRODUCTION 

It  is  well-known  that  the  occurrence  of  nonlinearities  in  engineering 
structures  can  have  a  significant  effect  on  their  behaviour.  The  most  spec¬ 
tacular  examples  of  this  can  be  found  in  the  literature  relating  to  chaotic 
systems;  the  response  of  such  a  system  to  a  deterministic  excitation  can 
be  unpredictable  beyond  a  short  time  scale.  Also,  any  stability  analysis 
for  a  nonlinear  system  will  depend  critically  on  the  type  of  nonlinearity 
present.  The  question  of  detecting  structural  nonlinearity  is  therefore  of 
some  importance. 

The  Hilbert  transform  is  now  established  as  a  means  of  diagnosing 
structural  nonlinearity  on  the  basis  of  measured  Frequency  Response  Func¬ 
tion  (FRF)  data  [8].  It  is  essentially  a  mapping  on  the  FRF  G{to), 

n[Gi.)]  =  Giu)^-yy§^  (1) 

(where  the  integral  is  to  be  understood  as  a  principal  value).  This  mapping 
reduces  to  the  identity  on  those  functions  corresponding  to  linear  systems. 
For  nonlinear  systems,  the  Hilbert  transform  results  in  a  distorted  version 
G,  of  the  original  FRF,  with  the  form  of  the  distortion  often  yielding  some 
indication  of  the  type  of  nonlinearity. 


1669 


The  origin  of  the  distortion  is  well-known  [10];  suppose  is  decom¬ 
posed  so, 

G{u>)  =  +  G  {u)  (2) 

where  G^{uj)  (resp.  G~{uj))  has  poles  only  in  the  upper  (resp.  lower)  half 
of  the  complex  w-plane.  It  is  a  straightforward  exercise  in  the  calculus  of 
residues  [9],  to  show  that  (with  a;  on  the  real  line), 

_  1  r  ^ _ =  _i_  (3) 

—  iJj  —  p 

if  p  is  in  the  upper  half-plane,  and, 


i7:J~Qo{Q—p){Q>  —  uj)  uj  —  p 
if  p  is  in  the  lower-half  plane.  It  follows  that, 

nlG^iuj)]  =  ±G^{u;)  (5) 

The  distortion  suffered  in  passing  from  the  FRF  to  the  Hilbert  trans¬ 
form  will  then  be  given  by  the  simple  relation, 


AGiuj)  =  n[G{uj)]  -  G{u;)  =  -2G‘{iv)  (6) 

This  relation  has  already  yielded  some  interesting  results  [4]  and  is  the 
basis  of  the  technique  presented  in  this  paper.  The  advantages  of  the  new 
approach  over  the  standard  time-  or  frequency- domain  methods  will  now 
be  explained. 

One  of  the  major  problems  in  using  the  Hilbert  transform  on  FRF  data, 
occurs  when  non-baseband  (data  which  does  not  start  at  zero  frequency)  or 
band-limited  data  is  employed.  Practically  speaking,  all  data  will  fall  into 
one  of  these  categories  and  the  problem  of  neglecting  the  contribution  of 
the  ‘out  of  band’  data  always  exists.  Establishing  baseband  data  is  rarely 
a  problem  in  simulation  work,  but  can  be  a  problem  with  experimental 
testing.  For  example,  it  is  known  [5],  that  the  use  of  random  excitation 
for  nonlinear  systems  produces  a  FRF  which  is  invariant  under  the  Hilbert 
transform.  This  means  that  other  excitation  signals  must  be  used.  Most 
commonly,  a  stepped-sine  signal  is  used,  and  this  is  time-consuming  due  to 
the  need  to  reach  a  steady-state  condition  at  each  frequency  increment.  As 
a  consequence,  this  excitation  is  usually  applied  over  a  limited  frequency 
range,  say  to 

Unfortunately,  the  idealised  Hilbert  transform  of  (1),  is  an  integral  over 
a  doubly-infinite  frequency  range.  In  practice,  the  data  over  the  intervals 
(-00,  (jJrrvin)  and  oo)  will  be  unavailable-  By  making  use  of  the  parity 


1670 


(oddness  or  eveness)  of  the  real  and  imaginary  parts  of  the  FRF ,  the  Hilbert 
transform  can  be  cast  in  the  following  form, 


^G{uj) 


9  roo 

-  dO. 

TrJo 


QGiQ)  n 
n-a;2 


(7) 


2uj  r 
TT  Jo 


dQ, 


(8) 


and  now  the  integrals  are  over  the  range  0  to  oo;  data  will  be  missing  from 
the  intervals  (0,a;^in)  and  {u^max,  oo). 

The  problem  of  truncated  FRF  data  is  usually  overcome  by  adding 
correction  terms  to  the  Hilbert  transform  evaluated  from  uimin  fo  u^max- 
general,  two  corrections  are  needed,  one  for  each  of  the  missing  intervals. 

In  order  to  display  the  inconvenience  of  the  procedure,  a  review  of  the 
relevant  theory  follows.  It  will  later  be  shown  that  the  method  proposed  in 
this  paper  circumvents  the  problem  of  the  missing  intervals  i.e.  truncation 
of  the  data. 


CORRECTION  TERMS 

There  are  essentially  three  methods  of  correcting  Hilbert  transforms, 
they  will  be  described  below  in  order  of  complexity. 

The  Fei  Correction  Method 

This  approach  was  developed  by  Fei  [2]  for  baseband  data  and  is  based 
on  the  asymptotic  behaviour  of  the  FRFs  of  linear  systems.  The  form  of 
the  correction  term  is  entirely  dependent  on  the  FRF  type;  receptance,  mo¬ 
bility,  or  inertance.  As  each  of  the  correction  terms  is  similar  in  principle, 
only  the  term  for  mobility  will  be  described. 

The  general  form  of  the  mobility  function  for  a  linear  system  with 
proportional  damping  is, 


G(‘-)  =  E  ^ 

where:  Ak  is  the  complex  modal  amplitude  of  the  mode;  u>k  is  the 
undamped  natural  frequency  of  the  mode  and  (k  is  its  viscous  damping 
ratio.  By  assuming  that  the  damping  is  small  and  that  the  truncation 
frequency,  tOmax  is  much  higher  than  the  natural  frequency  of  the  highest 
mode,  equation  (9)  can  be  reduced  to  (for  uj  >  tUmax), 

=  (10) 


1671 


which  is  an  approximation  to  the  ‘out  of  band’  FRF.  This  term  is  purely 
imaginary  and  thus  provides  a  correction  for  the  real  part  of  the  Hilbert 
transform  via  equation  (7).  No  correction  term  is  applied  to  the  imaginary 
part  as  the  error  is  assumed  to  be  small  under  the  specified  conditions. 

The  actual  correction  is  the  integral  in  equation  (1)  over  the  interval 
oo).  Hence  the  correction  term,  denoted  C^{uj),  for  the  real  part  of 
the  Hilbert  transform  is, 


oo 


dO. 


fi5(G(n))  2  “  dCl 


which,  after  a  little  algebra  [2],  leads  to, 


TTU  \idJnax  —  w 


(11) 


(12) 


The  Haoui  Correction  Method 


The  second  correction  term  which  again,  caters  specifically  for  baseband 
data,  is  based  on  a  different  approach.  The  term  was  developed  by  Haoui 
[3],  and  unlike  the  Fei  correction  has  a  simple  expression  independent  of 
the  type  of  FRF  data  used.  The  correction  for  the  real  part  of  the  Hilbert 
transform  is, 


Wmas 


dQ 


a^G{n)) 

n2_a;2 


(13) 


The  analysis  proceeds  by  assuming  a  Taylor  expansion  for  G{u;)  about 
and  expanding  the  term  (1  —  using  the  binomial  theorem. 

If  it  is  assumed  that  cj^ax  is  not  close  to  a  resonance  so  that  the  slope 
dG{co)/duj  (and  higher  derivatives)  can  be  neglected,  a  straightforward  cal¬ 
culation  yields. 


C^{u;)  =  C^{0)  - 


^{G{co„ 


.)) 


■  + 


+  ... 


(14) 


where  G^(0)  is  estimated  from, 


G^(0)  =  3J(G(0))  - 


"Ulrna.s 

j  da 

0+e 


S(g(ll)) 

a 


(15) 


Using  the  same  approach,  the  correction  term  for  the  imaginary  part, 
denoted  by  Can  be  obtained, 


1672 


TT 


5 

O)  UJ  U} 

- h  ^ - H 


3u^3  '  5t^5 

’-'“'max  “  max 


+  ... 


(16) 


The  Simon  Correction  Method 


This  method  of  correction  was  proposed  by  Simon  [7];  it  allows  for  trun¬ 
cation  at  a  low  frequency,  Umiu,  and  a  high  frequency  It  is  therefore 

suitable  for  use  with  zoomed  data.  This  facility  makes  the  method  the  most 
versatile  of  the  three.  As  before,  it  is  based  on  the  behaviour  of  the  linear 
FRF,  say  equation  (9)  for  mobility  data.  Splitting  the  Hilbert  transform 
over  three  frequency  ranges;  and  (w^arjOo),  the 

truncation  errors  on  the  real  part  of  the  Hilbert  transform,  at  low 

frequency  and  the  now  familiar  at  high  frequency,  can  be  written 

as, 


B- 


V)  =  -f/ 


2  T”,„fi&(G(w)) 


dO- 


(17) 


and 


=  / 


do. 


as{G{u)) 

f22_u.2 


(18) 


If  the  damping  can  be  assumed  to  be  small,  then  rewriting  equations 
(17)  and  (19)  using  the  mobility  form  (9),  yields. 


Wmtn  Jif 


(19) 


and 


Evaluating  these  integrals  gives, 


(20) 


N 


(  (^max  +  UJk){}^k  - 
\(^max  ^fc)(^/s  d"  ^min^  ) 


\  (^  ^mtn/(^max  j 

The  values  of  the  modal  parameters  Ajt  and  Uk 
initial  modal  analysis. 


(21) 


are  obtained  from  an 


1673 


Summary 

None  of  the  three  correction  methods  can  claim  to  be  faultless;  trunca¬ 
tion  near  to  a  resonance  will  always  give  poor  results.  Considerable  care 
is  needed  to  obtain  satisfactory  results.  The  Fei  and  Haoui  techniques 
are  only  suitable  for  use  with  baseband  data  and  the  Simon  correction  re¬ 
quires  a  prior  modal  analysis.  The  following  section  proposes  an  approach 
to  the  Hilbert  transform  which  does  not  require  correction  terms  and  thus 
overcomes  these  problems. 

THE  RATIONAL  POLYNOMIAL  METHOD 

The  basis  of  this  approach  to  calculating  a  Hilbert  transform  is  to 
establish  the  position  of  the  FRF  poles  in  the  complex  plane  and  thus 
form  the  decomposition  (2).  This  is  achieved  by  formulating  a  Rational 
Polynomial  (RP)  model  of  the  FRF  over  the  chosen  frequency  range  and 
then  converting  this  into  the  required  form  via  a  pole-zero  decomposition. 

A  general  FRF  may  be  expressed  as  the  sum  of  a  number  of  modes, 
e.g.  equation  (9)  for  the  mobility  form.  This  ’summation’  form  is  readily 
expanded  into  a  rational  polynomial  representation, 

GM  =  Ig  (22) 

where. 


TIQ  Tip 

iz=0  i=0 

and  the  polynomial  coefficients  and  &»,  are  functions  of  the  natural  fre¬ 
quencies,  dampings  and  Ak  of  the  modes.  Modal  analysis  methods  in  the 
frequency-domain  assume  either  a  summation  form  (9)  or  an  RP  form  (22). 
The  coefficients  are  determined  using  optimisation  schemes  which  find  the 
parameters  which  best  fit  the  data.  Various  methods  exist  which  max¬ 
imise  the  numerical  conditioning  and  precision  of  the  resulting  coefficients. 
The  RP  models  fitted  for  the  current  work  use  an  ‘instrumental  variable’ 
approach  which  minimises  the  effects  of  noise  and  produces  unbiased  coef¬ 
ficients.  The  details  of  the  method  can  be  found  in  [1]. 

Once  the  RP  model  Grp  is  established,  it  can  be  converted  into  a 
pole-zero  form, 


n 

GrpH  =  -  (24) 

-Pi) 


1674 


where  qi  and  pi  are  the  (complex)  zeroes  and  poles  respectively  of  the 
function.  The  next  stage  is  a  long  division  and  partial-fraction  analysis  in 
order  to  produce  the  decomposition  (2).  If  p^  (resp.  p~)  are  the  poles  in 
the  upper  (resp.  lower)  half-plane,  then, 


N+ 

G'^p{u))  =  ^ 


Gpp(u})  = 


cr 


(25) 


where  and  C~  are  coefficients  fixed  by  the  partial  fraction  analysis,  N+ 
(resp.  iV_)  is  the  number  of  poles  in  the  upper  (resp.  lower)  half-plane. 
Once  this  decomposition  is  established,  the  Hilbert  transform  follows  from 
(6).  (It  is  assumed  here  that  the  RP  model  has  more  poles  than  zeros. 
If  this  is  not  the  case,  the  decomposition  (2)  is  supplemented  by  a  term 
G°{uj)  which  is  analytic.  Equation  (6)  which  is  used  to  compute  the  Hilbert 
transform  is  unchanged.) 

The  procedure  described  above  is  demonstrated  in  the  following  section 
using  data  from  numerical  simulation. 


SIMULATION  DATA 


A  simulated  Duffing  oscillator  system  with  equation  of  motion, 

y  +  20y  +  lOOOOy  -f  5  x  10^/  =  X  (26) 

was  chosen  to  test  the  calculation  technique.  Data  was  generated  over  256 
spectral  lines  from  0  to  38.4  Hz  in  a  simulated  stepped-sine  test  based  on 
a  standard  fourth-order  Runge-Kutta  scheme  [6].  The  data  was  truncated 
by  removing  data  above  and  below  the  resonance  leaving  151  spectral  lines 
in  the  range  9.25-32.95  Hz. 

Two  simulations  were  carried  out.  In  the  first,  the  Duffing  oscillator 
was  excited  with  X  =  1.0  N  giving  a  change  in  the  resonant  frequency 
from  the  linear  conditions  of  15.9  Hz  to  16.35  Hz  and  in  amplitude  from 
503.24x10“®  m/N  to  483.0x10"®,  m/N.  The  FRF  Bode  plot  is  shown 
in  Figure  1,  the  cursor  lines  indicate  the  range  of  the  FRF  which  was 
used.  The  second  simulation  took  X  =  2.5  N  which  was  a  high  enough 
to  produce  a  jump  bifurcation  in  the  FRF.  In  this  case  the  maximum 
amplitude  of  401.26x10"®  m/N  occurred  at  a  frequency  of  19.75  Hz.  Note 
that  in  the  case  of  this  nonlinear  system  the  term  ‘resonance’  is  being  used 
to  indicate  the  position  of  maximum  gain  in  the  FRF. 

RESULTS 


The  first  stage  in  the  calculation  process  is  to  establish  the  RP  model 
of  the  FRF  data.  On  the  first  data  set  with  AT  =  1,  in  order  to  obtain  an 
accurate  model  of  the  FRF,  24  denominator  terms  and  25  numerator  terms 


1675 


were  used.  The  number  of  terms  in  the  polynomial  required  to  provide  an 
accurate  model  of  the  FRF  will  depend  on  several  factors  including:  the 
number  of  modes  in  the  frequency  range,  the  level  of  distortion  in  the  data 
and  the  amount  of  noise  present.  The  accuracy  of  the  RP  model  is  evident 
from  Figure  2  which  shows  a  Nyquist  plot  of  the  original  FRF,  G[(jj)  with 
the  model  Grp{uj)  overlaid  on  the  frequency  range  10  -  30  Hz. 

The  next  stage  in  the  calculation  is  to  obtain  the  pole-zero  decompo¬ 
sition  (24).  This  was  accomplished  here  by  solving  the  numerator  and 
denominator  polynomials  using  a  computer  algebra  package.  The  zeros 
and  poles  are  given  in  Tables  1  and  2  respectively.  It  may  not  be  nec¬ 
essary  to  obtain  the  zeroes  of  the  model,  this  is  under  investigation.  In 
order  to  check  the  stability  of  the  root  extraction,  the  pole-zero  model  was 
compared  with  the  measured  data,  an  exact  overlay  was  obtained. 

The  penultimate  stage  of  the  procedure  is  to  establish  the  decompo¬ 
sition  (2).  Given  the  pole-zero  form  of  the  model,  the  individual  pole 
contributions  are  obtained  by  carrying  out  a  partial  fraction  decomposi¬ 
tion,  because  of  the  complexity  of  the  model,  a  computer  algebra  was  again 
used. 

Finally,  the  Hilbert  transform  is  obtained  by  flipping  the  sign  of 
the  sum  of  the  pole  terms  in  the  lower  half-plane.  The  result  of  ths  calcu¬ 
lation  for  the  low  excitation  data  is  shown  in  Figure  3  in  a  Bode  amplitude 
format.  The  overlay  of  the  original  FRF  data  and  the  Hilbert  transform 
calculated  by  the  RP  method  is  given,  the  frequency  range  has  been  limited 
to  10  -  30  Hz. 

A  simple  test  of  the  accuracy  of  the  RP  Hilbert  transform  was  carried 
out.  A  Hilbert  transform  of  the  low  excitation  data  was  calculated  using 
a  standard  FFT  based  technique  [5]  on  an  FRF  using  a  range  of  0-50  Hz 
in  order  to  minimise  truncation  errors  in  the  calculation.  Figure  4  shows 
an  overlay  of  the  RP  Hilbert  transform  (from  the  truncated  data)  with 
that  calculated  from  the  FFT  technique.  The  Nyquist  format  is  used.  The 
figure  shows  that  a  satisfactory  level  of  agreement  has  been  achieved  with 
the  new  calculation  method. 

The  second,  high  excitation,  FRF  used  to  validate  the  approach  con¬ 
tained  a  bifurcation  or  ‘jump’  and  thus  offered  a  more  stringent  test  of  the 
RP  curve-fitter.  A  greater  number  of  terms  in  the  RP  model  were  required 
to  match  the  FRF.  Figure  5  shows  the  overlay  achieved  using  32  terms  in 
the  denominator  and  33  terms  in  the  numerator.  There  is  no  discernable 
difference.  Following  the  same  calculation  process  as  above  leads  to  the 
Hilbert  transform  shown  in  Figure  6,  shown  with  the  FRF. 

CONCLUSIONS 

A  method  of  calculating  Hilbert  transforms  from  truncated  Frequency 
Response  Function  (FRF)  data  is  presented  which  has  no  need  for  cor- 


1676 


rection  terms.  The  rational  polynomial  (RP)  technique  operates  without 
needing  the  functional  form  of  the  FRF  data,  it  applies  equally  well  to 
receptance,  mobility  and  inertance  forms. 

The  main  limitation  in  the  method  is  in  the  ability  of  the  RP  curve-fitter 
to  match  the  measured  FRF.  Sensitivity  of  the  method  to  noise  must  be 
investigated.  A  further  problem  relates  to  uniqueness  and  overfitting  of  the 
model.  This  has  already  been  addressed  in  the  context  of  modal  analysis 
e.g.  through  the  use  of  stability  plots,  and  the  solutions  discovered  there 
should  apply. 

The  implementation  of  the  algorithm  used  here  was  based  on  C  code 
and  computer  algebra.  It  is  intended  to  produce  a  single  unified  program 
to  carry  out  the  various  steps. 

ACKNOWLEDGEMENTS 

The  authors  would  like  to  thank  Dr.  Pete  Emmett  for  providing  the 
rational  polynomial  curve-fits  used  in  this  work. 


References 

[1]  Emmett  (P.R.)  1994  Ph.D  Thesis,  Department  of  Mechanical  Engi¬ 
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[2]  Fei  (B.J.)  1984  Rapport  de  stage  de  fin  d’ etudes,  ISMCM,  St  Ouen, 
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[3]  Haoui  (A.)  1984.  These  de  Docteur  Inginieur,  ISMCM,  St  Ouen, 
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University  of  Manchester.  Developments  in  the  modal  analysis  of  lin¬ 
ear  and  non-linear  structures. 


1677 


[8]  Simon  (M.)  &  Tomlinson  (G.R.)  1984  Journal  of  Sound  and  Vibration 
90  pp. 275-282.  Application  of  the  Hilbert  transform  in  modal  analysis 
of  linear  and  non-linear  structures. 

[9]  Stewart  (I.)  Tall  (D.)  1992  Complex  Analysis  Cambridge  University 
Press. 

[10]  Worden  (K.)  &  Tomlinson  (G.R.)  1990  Proceedings  of  the  8*^  Interna¬ 
tional  Modal  Analysis  Conference,  Orlando,  Florida  pp.  121-127.  The 
high-frequency  behaviour  of  frequency  response  functions  and  its  effect 
on  their  Hilbert  transforms. 


1678 


Numerator  Roots 


-1214.211008298219  -  726.1763609509598i 
-198.1314154141764  +  30.36502191203071i 
-183.7196601591438  +  0.07476628997601653i 
-173.0468329728469  -  0.58231241517227382 
-167.4159433485794  +  0.24993126862275962 
-148.7724514912166  -  0.5776974464158378i 
-106.029245566151  +  6.329765806548005i 
-105.755520052376  -  6.4400713086431752 
-103.9415121027209  -  0.22926381395787662 
-100.8868331667942  +  13.96371388356606i 
-100.3491120108979  -  13.55839223906875i 
-65.16772266742481  -  0.01242815581475099i 
-1.161302705005027  x  10“^°  +  1357.5723892932912 
65.16772266742481  -  0.012428155814750992 
100.3491120108979  -  13.55839223906875i 
100.8868331667942  +  13.96371388356606i 
103.9415121027209  -  0.2292638139578766i 
105.755520052376  -  6.440071308643175i 
106.029245566151  +  6.3297658065480052 
148.7724514912166  -  0.5776974464158373i 
167.4159433485794  +  0.2499312686227509i 
173.0468329728468  -  0.5823124151722609i 
183.7196601591439  +  0.07476628997601194i 
198.1314154141764  +  30.365021912030712 
1214.211008298219  -  726.1763609509598i 


Table  1:  Polynomial  Numerator  Roots  for  HCSl  data 


1679 


_ Denominator  Roots _ 

-198.1567099579465  +  30.40483782876573i 
-183.7196564899432  +  0.07470973423345939i 
-173.0468138842385  -  0.58222324003507392 
-167.4160993615824  +  0.24976877099244062* 
-148.7725565604598  -  0.5774140178292062* 
-106.1570440779208  -  5.3529817854685192 
-105.7091945278726  +  6.502652089082012 
-103.941962628465  -  0.2315241290997767i 
-102.9807367369542  -  9.760731644734012 
-100.121488863284  +  14.02519387639362 
-97.6024028604877  -  15.116504458164732 
-65.16768960663572  -  0.012732667738438052 
65.16768960663572  -  0.01273266773843805i 
97.6024028604877  -  15.116504458164732 
100.121488863284  +  14.02519387639362 
102.9807367369542  -  9.760731644734012 
103.941962628465  -  0.2315241290997704i 
105.7091945278726  +  6.502652089082006i 
106.1570440779208  -  5.3529817854685232 
148.7725565604598  -  0.57741401782920622 
167.4160993615824  -I-  0.24976877099244132 
173.0468138842385  -  0.5822232400350746i 
183.7196564899432  +  0.07470973423345935i 
198.1567099579465  +  30.40483782876573i 


Table  2:  Polynomial  Denominator  Roots  for  HCSl  data 


1680 


0.0000  Frequency  /  Hz  38.400 


Figure  1:  Bode  plot  of  low  exitation  Duffing  oscillator  FRF. 


J _ ! _ ! _ 1 - 1 - 1— - 1 - ! - ^ 

-500. 00U  Real  /  mjN  +300. 00u 


Figure  2:  Overlay  of  RP  model  Grp{u})  and  original  FRF  for  low 

excitation  Duffing  oscillator. 


1681 


Figure  3:  Original  G{uj)  and  RP  Hilbert  transform  Grp(u}):  Duffing  oscil¬ 
lator  -  low  excitation. 


Figure  4:  Comparison  of  RT  and  FFT  Hilbert  transforms  for  low  excitation 
Duffing  system. 


1682 


Figure  5;  Overlay  of  RP  model  Grp{uj)  and  original  FRF  G{oj)  for  high 
excitation  Duffing  oscillator. 


Figure  6:  Original  G(w)  and  RP  Hilbert  transform  Grp{u}):  Duffing  oscil¬ 
lator  -  low  excitation. 


1684 


Fractional  Fourier  Transforms  and  their  Interpretation 
D.M.  Lopes,  J.K.  Hammond  and  P.R. White 


Institute  of  Sound  and  Vibration  Research 
University  of  Southampton, 
Southampton  SOI  7-1 BJ 
United  Kingdom 


Abstract 

Fractional  Fourier  Transforms  arise  naturally  from  optical  signal 
processing.  However,  their  interpretation  for  one-dimensional  signals 
are  still  to  be  fully  explored.  This  paper  is  concerned  with  fundamental 
aspects,  including : 

•  Definition  and  relation  to  optics 

•  Interpretation  of  signal  decomposed  in  terms  of  'linear  chirps' 

•  A  generalised  concept  of  group  delay 

•  The  relationship  to  time-frequency  distributions 

•  Analytic  and  computational  examples  showing  the  relevance 
of  the  decomposition  for  non-stationary  signals  occurring 
commonly  in  acoustics 


Introduction 

The  Fourier  Transform  (FT)  is  perhaps  the  most  widely  used 
tool  in  the  signal  processor's  armament.  It's  simple  definition,  both  in 
the  analogue  and  digital  domains  enables  fast  and  efficient 
computation.  However,  we  should  sometimes  consider  whether  the 
information  given  to  us  via  the  FT,  could  not  be  better  represented  in 
some  other  form.  The  Fractional  Fourier  Transform  (FRFT)  is  a 
generalisation  of  the  traditional  FT,  which  by  changing  a  given 
parameter  in  the  transform  allows  us  to  continuously  move  between 
the  time  and  frequency  domains.  First  used  for  solving  fractional 
differential  equations,  the  transform  later  found  use  in  optical  physics, 
and  this  link  is  explored  in  the  first  section.  It  can  be  shown  that  the 
FRFT  can  be  viewed  as  a  transform  which  has  a  set  of  basis  functions 
consisting  of  'linear  chirps';  this  is  covered  in  the  second  section.  These 
are  simple  harmonic  waves,  whose  frequency  changes  linearly  with 


1685 


time.  In  the  third  section  we  concern  ourselves  with  the  question  : 
'Given  a  signal,  at  what  time  does  linear  chirp  make  its  maximum 
contribution?'  This  is  a  generalisation  of  the  work  done  by  Zadeh  [9] 
for  the  traditional  Fourier  case.  We  then  investigate  the  link  between 
the  FRFT  and  Time-Frequency  Distributions  (TFD).  The  link  between 
the  FRFT  and  the  Wigner  Distribution  (WD)  is  explored.  We  complete 
the  paper  by  illustrating  the  above  work  using  an  analytical  and 
numerical  examples  of  the  manner  in  which  the  FRFT  decomposes 
signals. 


The  FRFT  and  its  Relation  to  Optical  Physics 

The  simplest  way  of  visualising  the  form  of  the  FRFT  is  to 
consider  it  in  terms  of  simple  optical  physics.  Figure  1  shows  such  an 
example  of  a  simple  projector  type  system.  Ozaktas  [5]  was  the  first  to 
view  the  FRFT  in  this  form,  and  started  transition  of  the  transform 
from  pure  mathematics  into  more  general  usage.  Even  from  this 
simple  system,  it  is  useful  to  define  a  few  key  points.  The  input  image 
into  the  system  can  be  considered  to  be  the  resultant  image  produce 
immediately  behind  the  lens,  labelled  a=0  on  the  figure.  As  we  move 
further  away  from  the  lens  the  image  converges  at  the  focal  plane,  this 
is  labelled  on  the  figure  as  a=;r/2  [1]. 

At  the  focal  plane  the  image  produced  is  the  2-D  Fourier 
transform  of  the  input  image.  If  the  input  image  is  constant  coherent 
light  across  the  whole  of  the  input  image  plane,  then  at  the  focal  plane 
we  would  expect  to  simply  see  a  point  of  light.  This  is  because  the 
Fourier  Transform  of  a  constant  is  a  delta  function,  as  in  this  case  we 
are  dealing  m  two  dimensions  it  therefore  follows  that  we  should  see  a 
'spot'  in  the  centre  of  the  screen. 

As  we  move  past  the  focal  plane  the  image  begins  to  expand 
once  more.  At  a  certain  distance  we  will  recover  a  scaled  version  of  the 
original  image,  this  point  is  labelled  0i=7r.  At  this  point  we  have 
applied  two  Fourier  transforms  to  the  input  image,  resulting  in  a  scaled 
version  of  the  input.  This  is  as  expected  from  the  theory  of  the  Fourier 
transform.  We  have  therefore  seen  that  the  Fourier  transform  of  the 
input  signal  is  at  a  distance  from  the  lens  of  a^n/2  and  that  the 
operation  is  repeated  at  a  distance  of  a-K.  At  distances  from  the  lens, 
which  are  not  multiples  of  k/2,  the  resultant  image  is  a  fractional 
Fourier  Transform  of  the  input  image. 

These  optical  properties  are  stated  in  mathematical  terms  in  [2- 
3].  A  given  transform  (F“)  can  be  considered  to  be  a  FRFT,  if  it 


1686 


conforms  to  the  a  set  of  properties  as  outlined  in  Almeida's  paper  [4]. 
(Where  I  is  the  identity  operator) 


1.  Zero  Rotation  :  I 

2.  Consistency  with  the  Fourier  Transform  :  FT 

3.  Additivity  of  Transform : 

4.  Periodic  :  I 


Although  these  properties  define  the  FRFT  completely,  they  do 
not  uniquely  define  it.  However  it  has  been  shown  that  the  majority  of 
the  definitions  given  in  the  literature  are  consistent. 


Linear  Representation  of  the  FRFT 

Almeida  [4]  defined  such  a  FRFT  as  follows  : 

oo 

XJu)  =  J x{t)K^(t,u)dt  ^  x(t)  =  J X„{u)K.^{u,t)du  (1) 

Were  K.oit,u)  is  the  kernel  of  the  FRFT  as  is  defined  as  : 

ll-JcotCa) 

\  2k  ^ 

K^{t,u)  =  l  6(t-u)  a  =  2nK  neZ 

5(t  +  u)  a  +  K~2nK 


1687 


Where  t  is  the  time  variable,  and  u  is  the  generalised  frequency 
axis  (i.e.  (o  as  a-47u/2).  Almeida  approached  the  FRFT  in  much  the 
same  manner  as  other  authors  on  the  topic.  Starting  from  the  basic 
properties  of  the  FRFT,  he  expressed  the  FRFT  in  a  form  similar  to  that 
expressed  by  Ozaktas  [5].  Almeida  stated  that  the  FRFT  consists  of 
expressing  x(t}  on  a  basis  formed  by  the  set  of  functions  defined  by  the 
kernel  (u,t).  The  basis  is  orthonormal  since  it  can  be  shown  that : 

Bl{u  ,t)  dt  =  S{u-u!)  (2) 


The  basis  functions  are  'chirps',  i.e.  complex  exponentials  with 
linear  frequency  modulation.  The  FRFT  can  now  be  viewed  as  the 
decomposition  of  the  signal  in  terms  of  linearly  changing  harmonic 
waves  or  'chirps',  sweeping  at  a  rate  dependant  upon  the  parameter  a. 
This  can  be  seen  to  be  consistent  with  the  definition  of  the  FRFT  as  a 
generalised  FT,  since  in  the  limit  as  the  parameter  a-»7iy2,  the  linear 
chirps  become  simple  harmonic  waves. 

We  also  can  consider  the  FRFT  to  be  a  rotation  of  the  TF  plane. 
If  a=0  then  we  are  along  the  time  axis  and  thus  the  input  signal 
remains  imchanged.  If  a=;z/2,  then  we  are  on  the  frequency  axis,  and 
thus  the  FRFT  should  be  precisely  the  same  as  the  FT.  At  rotations 
which  are  not  multiple  of  ;r/2,  we  are  then  in  a  domain  which  is  neither 
time  nor  frequency  but  a  mixture  of  both. 

The  parameter  a  can  be  considered  to  be  the  chirp  rate,  that  is 
the  rate  at  which  the  swept  sine  waves  of  the  basis  functions  increase  in 
frequency.  As  a^7i/2  the  basis  functions  revert  to  the  simple  harmonic 
waves,  as  a  reduces  down  to  zero,  the  rate  at  which  the  chirps  change 
in  frequency  increases.  It  can  be  shown  that  as  a-^0  that  a  chirp  in 
Eqn.  1  can  be  viewed  as  a  Delta  function  [10,  p.99].  The  definition  of 
the  transform  as  written  in  terms  of  the  kernel  function  as  defined  by 
Almeida  and  Ozaktas  then  becomes  consistent  with  the  viewing  the 
FRFT  as  a  rotation  of  the  FT  plane. 

Almeida's  linear  representation  of  the  FRFT  as  a  decomposition 
of  a  signal  in  terms  of  linearly  modulated  harmoiuc  waves  allows  an 
intuitive  hold  upon  the  transform,  and  can  be  seen  to  be  of  vital 
importance  over  the  next  few  pages. 


1688 


The  Generalised  Concept  of  Group  Delay 

The  idea  of  Group  Delay  (GD)  was  well  explained  by  Zadeh,  [9, 
p.  441-444]  the  core  of  his  idea  is  as  follows.  In  it's  traditional 
definition  GD  is  the  amount  by  which  a  given  (very  small)  band  of 
frequencies  make  their  maximum  contribution  towards  a  signal. 
Zadeh  was  able  to  show  that  the  GD  of  a  given  signal  could  be 
expressed  as  follows  : 


A  ^  sin 


cos{coj-^{co„„)) 


where  A  is  the  bandwidth  of  the  group  of  frequencies  under 

consideration,  and  0'  =  — —  .  This  can  be  viewed  as  a  sum  of  cosine 

dO) 

waves,  modulated  by  a  sinc(x)  type  function,  which  has  its  centre  (and 
maximum  value)  at  -  q' .  This  means  that  by  finding  0'  we  can  find  the 
time  at  which  a  band  of  frequencies  make  their  maximum  contribution 
to  the  signal. 

A  classical  example  of  group  delay  can  be  seen  if  we  introduce  a 
pure  delay  into  the  input  signal.  A  pure  delay  of  a  signal  introduces  a 
phase  shift  in  the  FT  as  follows  : 

The  complex  exponential  does  not  change  the  amplitude  of  the 
FT,  but  introduces  a  linearly  changing  phase  whose  negative  gradient, 
-0',  is  proportional  to  the  delay  introduced  into  the  signal.  It  thus 
follows  that  the  derivative  of  the  phase,  -0',  is  constant,  and  is  the 
amount  by  which  the  signal  has  been  delayed.  This  serves  to  reaffirm 
the  definition  of  group  delay,  since  we  defined  -0'  as  the  time  at 
which  a  band  of  frequencies  make  their  maximum  contribution,  which 
in  this  case  is  the  amount  by  which  the  signal  has  been  delayed. 

We  now  show  that  is  it  possible  to  extend  Zadeh's  original 
definition  using  the  FRFT,  to  answer  the  question  :  At  what  time  does  a 
'chirp'  of  rate  a,  with  centre  along  the  u  axis  make  it's  maximum 
contribution  to  the  signal?  Starting  from  our  original  definition  of 

taking  the  inverse  transform  back  into  the  time  domain  and  writing 
the  transform  in  terms  of  an  modulus  and  phase  it  follows  that : 


1689 


- cot(a)-a;csc(a)-0„(w) 


2k 


du 


By  dividing  up  the  u  axis  into  equal  segments  of  length  A.  Then 


(3) 


x(t)  = 


l-ycot(a) 


(n+I)A 


2k 


X  J  4(“)-« 


— — cot(a)-Kfcsc(a)-0„(«) 


du  (4) 


n=-oo 


If  we  make  A  sufficiently  small  so  that  for  all  segments  Aa  is 
constant,  we  can  make  the  following  re-arrangement,  defining  the 

centre  of  the  bandwidth  as  +-^jA : 


l-Jcot(a) 

'“■’I— s— 


(n+l)A 


X  j 

n=-oo  ^ 


j\ - co[(Qr)+///csc(a)-(>„(i;) 


du  (5) 


Furthermore  if  we  require  that  A  be  sufficiently  small  that  over 
each  internal  nA..(n+I)A  we  can  use  the  Taylor  expansion  for  (Pq(^)^ 
centred  around  giving 


(l)„iu)  =  (l)^(u^J  +  (u-u„J 


du 


By  defining  <})■  =  ,  and  making  the  above  substitution  into  Eqn.  5 


we  can  write 


l-Jcot(a) 


X  J  ^ 


-  cot(  a  )+ut  csc(a  )-0„  ( u„„ )-( u~u„„  )0’ 


du  (6) 


n=—oo 


After  a  little  manipulation  of  the  exponential  it  is  possible  to  show  : 

1  _  /cotfa)  -y[  ^cot(a)+0„{u„„ )-«„„/ csc(a)  "»»!’^  y— !^cot(a)+(fcsc(a)-6')(«-«, 

LK  n=-oo 

(7) 

The  integral  can  be  expressed  in  terms  of  Error  functions,  and  it 
is  then  possible  to  write  Eqn.  7  as  follows: 


1690 


^(0  = 


/l-;’cot(ay  42 


It:  y  J  cot(a) 


n=-oo 


_J|r|^^i^l!^+csc-(a)j+f(-u„„c^*(a)-2u„„csc(a)a)l(a)-20\*sc(a>)-^>,,{n,,J+2(5'w„„c(^t(a)+(^'- Erf[l]] 

(8) 

Where  Erf  [1],  [2]  are  defined  as  : 

'  J 42  {-2u„„  cot(a)  -  A  cotfa)  +  2t  csc(a)  -  20' )  ^ 


Erfm  =  Erf 


Erf  [2]  =  Erf 


4^;cot(a) 

7V2(-2m„„  cot(a)  + Acot(Q:)-H2rcsc(a)-20') 
44Jcot(a) 


) 


Although  somewhat  complicated  the  result  can  be  considered  to 
be  a  generalisation  of  Zadeh^s  result,  as  a-^7c/2,  the  above  result 
simplifies  down  to  Zadeh^s  result.  Loosely  the  Erf  terms  can  be 
considered  to  be  the  general  form  of  the  sine  function  used  in  the 
traditional  Fourier  case.  The  parameters  u^n  and  b'  shift  the  centre  of 
the  function,  A  effects  bandwidth,  and  a  tightens  the  function  about 
the  centre.  As  with  the  Zadeh  definition,  we  now  wish  to  find  the 
maximum  value  of  the  modulating  function,  to  determine  when  the 
chirp  makes  it's  maximum  contribution.  By  elementary  mathematics 
this  can  be  found  as  : 

t  =  cos(a)  +  0'  sin(a))  (9) 


This  can  be  checked  against  Zadeh's  result  by  setting  the  sweep 
rate  to  a=%/2.  The  result  is  interesting,  because  of  the  underlining 
simplicity  hiding  the  complex  mathematics  preceding  it.  Stated  again, 
this  can  be  seen  as  the  maximum  time  at  which  a  linearly  swept  sine 
wave  makes  its  maximum  contribution  to  a  given  signal. 


The  FRFT  and  it's  Relation  to  TFD's 

The  concept  of  GD  (and  instantaneous  frequency)  allows  us  to 
view  the  internal  structure  of  analytical  signals.  However,  for  non¬ 
stationary  or  time-varying  signal  we  need  some  other  method  to  view 


1691 


the  changing  frequency  components  of  a  signal  (i.e.  [1]).  The  use  of 
Time-Frequency  analysis  allows  us  to  construct  a  TF  plane,  allowing 
the  visualisation  of  the  signal  in  terms  of  time  and  frequency. 
Almeida's  most  significant  contribution  to  the  field  of  FRFT  was  to  link 
the  FRFT  and  Time-Frequency  analysis.  Almeida  viewed  the  FRFT  in 
terms  of  a  rotation  of  the  Time-Frequency  Distribution  (TFD)  and  the 
application  of  a  Fourier  Transform.  More  important  still  was  the  link 
that  Almeida  established  between  the  Wigner  Distribution  (WD)  which 
is  one  such  method  of  constructing  the  TFD  and  the  FRFT. 

After  a  little  mathematics  (see  [4])  Almeida's  deduced  that  the 
WD  could  be  written  as  ; 


Where  u  =  t  cos  a  +  co  a ,  and  v  =  -  ?  a  +  ©  cos  a 

The  left  hand  side  of  this  equation  is  the  WD  of  Xo[ u)  computed 
with  the  arguments  (r,©),  and  the  right  is  the  WD  of  x(t)  computed  with 
arguments  (m,v).  The  equation  shows  that  the  WD  of  Xa  is  the  same  as 
the  WD  of  X  if  we  take  into  account  the  rotation  due  to  the  change  of 
variables. 

"  This  is  equivalent  to  saying  that  the  WD  of  Xa  is  the  WD  of  x, 
rotated  by  an  angle  of  -a,  or  that  it  is  simply  the  WD  of  x  expressed  in 
the  new  set  of  co-ordinates  Almeida,  [4,  p.  3088] 

Lohmarm  and  Soffer's  paper  [6]  shows  essentially  the  same 
property,  but  use  the  Radon  transform  to  produce  the  required 
rotation  and  projection.  Since  Almeida  published  his  paper,  other 
authors  have  established  this  link  not  just  for  the  WD,  Ozaktas  et  al.  [7] 
showed  that  the  same  properties  were  true  for  all  members  of  the 
Cohen  class  which  are  rotationally  symmetric  about  the  origin. 

Another  way  of  viewing  the  FRFT  is  as  a  rotation  and  projection 
of  the  TF  plane.  Figure  2  shows  this  idea.  We  can  assume  that  a  linear 
chirp  is  simply  a  straight  line  in  the  TF  plane.  If  we  rotate  the  time  and 
frequency  axis  by  the  sweep  rate  of  the  chirp,  and  project  along  the  u 
axis,  we  have  constructed  the  FRFT.  This  is  an  intuitive  representation 
of  the  mathematics  given  by  Almeida  and  Ozatkas.  This  rotation  and 
projection  is  called  the  Radon  transform,  and  in  [6]  the  presentation  of 
the  FRFT  as  a  Radon  transformed  WD  was  shown. 


1692 


Figure  2  -  FRFT  is  terms  of  rotation  and  projection. 


Frequency 


Examples 


1.  Rectangular  Function 

The  first  of  our  examples,  shows  the  analytical  decomposition 
for  a  rectangular  function.  The  FRFT  can  be  written  in  terms  of  Fresnel 
integrals,  as  expressed  below  : 


’  2cot(a) 


y 

2 


(‘■4) 

K 

(  2b|5-r^ 

-K 

^2ap~y^ 

[  VW  ; 

l^/W  JJ 

(11) 


Where  K  is  defined  as  the  Fresnel  integral : 

X  jm~ 

K{x)  =  \e  -  dz  (12) 

0 

This  function  is  shown  in  Figure  3.  The  function  can  be  seen  to 
evolve  as  the  sweep  rate  is  changed.  As  a— >0  we  can  seen  that  the 
FRFT  tends  towards  the  rectangular  function  as  expected.  As  the 
sweep  rate  increases  the  function  develops,  with  oscillations  in  the 
transforms  becoming  slower.  As  a“»7c/2  the  function  develops  into  the 
a  sine  type  function,  the  Fourier  transform  of  a  rectangle. 


1693 


2.  Doppler  Type  Signals 

The  acoustics  of  a  moving  sound  source  are  extremely 
complicated,  both  the  shift  in  frequency  (Doppler  shift)  due  to  the 
movement  of  the  source,  and  the  propagation  delay  of  the  speed  of 
soxmd  must  be  taken  into  accoxmt.  In  the  following  example,  we 
simplify  the  situation  by  considering  the  Doppler  shift  without 
propagation  delay.  In  order  to  simulate  the  change  in  level  of  the 
sound  experienced  due  to  the  distance  of  the  vehicle  from  the 
microphone,  we  will  use  a  Gaussian  function  to  reduce  the  signal  to 
zero  at  the  beginning  and  end  of  the  signal. 

If  the  source  and  receiver  approach  each  other  at  speeds  of  v 
and  u  respectively,  and  the  source  emits  a  tome  frequency  of  f„  when 
stationary,  then  it  is  possible  to  show  ([8],  p.  415)  that  the  receiver  will 
hear  a  frequency/’  where  :  (c  is  the  speed  of  sound) 


/  =  /o—  (13) 

c-v 

This  change  in  frequency  is  known  as  the  Doppler  Shift.  A  schematic  of 
the  type  of  system  we  are  trying  to  model  is  shown  in  Figure  4. 

If  the  microphone  is  stationary,  and  the  car  moves  passed  it  at 
constant  speed,  then  a  modified  form  of  the  equations  [8]  describes  the 
frequency  received  by  the  microphone  (the  signal  is  shifted  in  time  in 
order  to  keep  the  equations  causal)  : 


f  =  fo 


1  +  - 


VCOSI 


m' 


(14) 


1694 


were  0(f)  = 


-tan 


;r-tan 


v(f-lO) 


v(r-lO) 


f  <  10 
f>10 


This  is  a  non-linear  function;  however  as  the  car  approaches  the 
microphone,  it  becomes  almost  linear.  Figure  5  shows  the  changing 
frequency  component  with  time  for  two  different  (simulated)  vehicle 
speeds.  (The  curve  for  the  faster  speed  is  marked  with  signs.)  We 
will  assume  for  the  sake  of  simplicity,  that  the  vehicle  emits  only  one 
tone,  and  that  it  travels  at  constant  velocity. 


1695 


The  greater  the  velocity  of  the  vehicle,  the  steeper  the  curve. 
The  angle,  Q,  varies  from  Q..k.  The  rate  of  decrease  of  frequency  can  be 
found,  and  is  shown  to  be 


dt 


xL  sin 


tan' 


v(r-lO) 


c{t-\0)- 


1  +  — 


v'(r-lO)' 


(15) 


Since  the  rate  of  change  of  frequency  is  almost  linear  at  the  point 
when  the  vehicle  passes  the  microphone,  application  of  the  FRFT  (at 
the  correct  sweep  rate  setting)  will  produce  a  strong  spike  like 
distribution.  From  the  knowledge  of  the  sweep  rate  and  thus  the  rate 
of  change  of  frequency,  it  is  possible  to  infer  the  velocity  of  the  vehicle. 
Using  this  simple  model  of  a  Doppler  shifted  signal,  and  making  some 
basic  assumptions  (the  observation  point  i.e.  microphone  is  stationary, 
constant  velocity,  and  single  pure  tone  emission  of  the  source)  we 
simulate  the  situation.  The  synthesised  signal  is  shown  in  Figure  6, 
and  its  Spectrogram  is  depicted  in  Figure  7. 

From  the  synthetic  signal  the  FRFT  is  used  to  find  the  sweep 
rate  which  best  matches  that  of  the  input.  Thus  is  we  pick  the  right 
sweep  rate,  the  chirp  produced  by  the  vehicle  will  show  up  as  a  spike, 
or  an  least  a  peak  in  the  distribution.  In  our  example  case  were  we 
know  the  input  parameters, 


1696 


it  is  possible  to  predict  at  what  sweep  rate,  and  were  on  the  u  domain 
we  expect  to  see  a  rise  in  the  distribution.  In  this  example  case  it  has 
been  possible  to  predict  the  sweep  rate  and  position  of  peak  on  the 
axis,  however  in  a  real  situation  this  would  not  be  the  case.  Multiple 
transforms  would  have  to  be  taken  on  the  data,  possibly  in  some 
iterative  way  to  find  the  peak  in  the  distribution.  However  due  to  the 
nature  of  the  delay  equations,  and  the  low  sweep  rate,  the  location  of 
the  peak  will  not  change  very  much  from  one  velocity  to  another. 

The  result  of  the  application  of  the  FRFT  can  be  seen  in  Figure  8, 
the  peak  which  was  expected  does  not  stand  out  greatly  from  the  rest 
of  the  data  because  of  the  way  in  which  the  signal  changes.  Close  to 
the  region  we  are  considering  to  be  a  linear  chirp,  other  almost  linear 
chirps  are  also  present.  These  other  chirps  spread  out  the  distribution 
into  the  form  seen  in  Figure  8. 

From  the  knowledge  of  the  rate  of  change  of  frequency,  it  is  a 
simple  manner  to  calculate  the  passing  velocity  of  the  vehicle. 


1697 


Frequency 


Figure  7  -  TF  Plane  of  input  signal. 
Sin(Phi) 


Time 


1698 


Conclusions 


In  this  paper  we  have  covered  some  of  the  theory  and  applications  of 
the  fractional  Fourier  transform.  We  have  shown  that  the  fractional 
Fourier  transform  can  be  considered  to  be  a  generalised  version  of  the 
Fourier  transform,  having  a  basis  of  swept  sine  waves  rather  than 
harmonic  waves.  We  have  considered  the  notion  of  ^group  delay'  as 
being  the  time  and  which  a  band  of  frequencies,  and  generalised  this 
idea  to  the  form  where  by  we  can  answer  the  question  'At  what  time 
does  a  chirp,  of  a  given  sweep  rate  make  it's  maximum  contribution  to 
a  signal  ?'  The  link  between  the  FRFT  and  TF  has  been  explored.  We 
have  investigated  some  examples  of  signal  using  this  techmque.  Finally 
we  have  outlined  one  application  of  the  FRFT  as  a  way  of  determining 
the  degree  of  shift  in  a  Doppler  type  signal. 

References 

1)  D.  Lopes  1996  Disseration  submitted  as  part  of  an  M.Sc.  in  Soimd 
and  Vibration,  at  ISVR,  University  of  Southampton. 

2)  A.C.  McBride  and  F.H.Kerr  1985  IMA  Journal  of  Applied  Mathematics 
39, 159-175.  On  Namias's  Fractional  Fourier  Transforms 

3)  S.Abe  and  J.T.Sheridan  1994  /.  Phys.  A:  Math.  Gen.  27,  4179-4187. 
Generalization  of  the  fractional  Fourier  transform  to  an  arbitrary  linear 
lossless  transform:  an  operator  approach 

4)  L.  B.  Almeida  1994  IEEE  Transactions  on  Signal  Processing  42(11), 
3084-3091.  The  Fractional  Fourier  Transform 

5)  FI.M.Ozaktas  and  B.Barshan  1994  /.  Opt.  Soc.  Am.  A.  11(2),  547-561. 
Colvolution,  filtering,  and  multiplexing  in  fractional  Fourier  domain 
and  their  relation  to  chirp  and  wavelet  transforms. 

6)  A.W.Lohmannm  and  B.H.Soffer  1995  J.  Opt.  Soc.  Am.  A  11(6),  1798- 
1801.  Relationships  between  the  Radon-Wigner  and  fractional  Fourier 
Transform 

7)  H.M.Ozaktas,  N.Erkaya  and  M.A.Kutah  1996  IEEE  Signal  Processing 
Letters  3(2),  40-41.Effect  of  Fractional  Fourier  Transformation  on  Time- 
Frequency  Distributions  Belonging  to  the  Cohen  Class 

8)  L.E.Kinsler  and  A.R.Frey  1982  Fundamentals  of  Acoustics.  New  York  : 
John  Wiley  and  Sons  Inc. 

9)  L.A.  Zadeh  and  C.Desoer  1963  Linear  System  Theory.  New  York  : 
McGraw-Hill. 

10)  A.  Papoulis  1984  Signal  Analysis.  New  York  :  McGraw-Hill. 


1699 


1700 


SYSTEM  IDENTIFICATION  III 


WAVE  LOCALIZATION  EFFECTS  IN  DYNAMIC  SYSTEMS 


J.  Dickey 

Center  for  Nondestructive  Evaluation, 

The  Johns  Hopkins  University,  Baltimore,  MD  21218,  USA 

G.  Maidanik 

David  Taylor  Research  Center.  Bethesda,  MD  20084,  USA 
J.  M.  D'Archangelo 

United  States  Naval  Academy.  Annapolis,  MD  21402,  USA 

Abstract 

Structures  with  discrete  periodic  variations  in  impedance  may  exhibit  pass 
and  stop  bands  and  the  related  wave  localization  and  delocalization  phenomena 
in  their  frequency  response.  Localization,  similar  to  Anderson  localization  in 
atomic  systems,  occurs  in  the  pass-band  frequency  range  when  the  periodicity 
is  perturbed  and  waves  are  thereby  inhibited  from  propagating.  Conversely, 
delocalization  occurs  in  the  same  systems  in  the  stop-band  regions  where 
perturbing  the  strict  periodicity  allows  for  relatively  more  propagation. 
Localization  and  delocalization  are  demonstrated  in  several  systems: 
specifically,  a  beaded  string,  membranes  and  plates  with  periodic  stiffeners 
attached,  and  a  "jungle  gym",  i.e.  a  connected  beam  structure.  It  is 
demonstrated  that  these  effects  depend  on  the  interactions  between 
discontinuities. 

Introduction 

A  structure  with  strictly  periodic  impedance  discontinuities  exhibits  pass  and 
stop  bands  in  its  frequency  response.  The  pass  bands  are  frequencies  at  which 
waves  traverse  the  structure  relatively  unimpeded  and  thus  give  rise  to  modes 
which  span  the  entire  structure:  i.e.  global.  Stop  bands  are  frequencies  at 
which  waves  are  impeded  from  traversing  the  structure  and  give  rise  to  modes 
which  are  confined  near  the  source:  i.e.  local.  When  the  strict  periodicity  of 
the  structure  is  disturbed  by.  e.g.,  randomly  varying  the  distances  (or 
equivalently,  the  wave  propagation  speed)  between  discontinuities,  the  pass- 


1701 


and  stop-band  character  in  the  frequency  response  is  ,  in  some  degree,  spoiled. 
The  "spoilage"  of  the  strict  pass-band  behavior  results  in  the  long  range  wave 
propagation  becoming  shortened  with  the  attendant  localization  of  the  modes. 
This  effect  is  called  localization.  Similarly,  the  "spoilage"  of  the  strict  stop- 
band  behavior  results  in  a  lengthening  of  the  wave  propagation  and  a 
delocalization  of  the  modes.  This  effect  is  called  delocalization.  Both 
localizing  and  delocalizing  cause  the  long  and  short  range  wave  propagation, 
respectively,  to  tend  towards  a  middle  ground,  called  a  "yellow"  band  (amber 
to  some),  where  waves  are  neither  short  nor  long  and  where  modes  are  neither 
global  nor  local.  It  is  proposed  that  delocalization  is  as  important  a  phenomena 
as  localization  with  relevance  to  practical  applications. 

These  phenomena  occur  in  a  wide  variety  of  dynamic  systems  ranging  from 
atomic  lattices  to  rib-stiffened  ship  hulls.^’-*^^  In  fact,  the  phenomenon  of 
localization  in  atomic  systems  was  popularized  by  P.  W.  Anderson  in  1958 
who  characterized  it  as  "absence  of  diffusion  in  certain  random  lattices 
Anderson  was  working  with  the  wave  equation  describing  the  electron  density 
in  crystalline  lattices  and  noted  that  randomizing  the  spacings  of  an  originally 
regularly  spaced  crystal  tended  to  inhibit  the  diffusion  of  electrons  from  a  point 
of  injection.  The  analysis  here  is  limited  to  macroscopic  systems  obeying 
linear  wave  equations  since  the  effects  are  fundamentally  phase  interference 
and.  as  such,  depend  on  superpositon  of  wave  functions.  This  is  not  to  say 
that  localization  does  not  exist  in  non-linear  systems,  it  is  Just  that  the  model 
used  to  describe  it  here  is  limited  to  linear  systems.  Further,  the  analysis  is 
limited  to  one-dimensional  systems  in  the  sense  that  the  wave  equation  for  the 
structure  must  be  separable  into  functions  of  a  single  spatial  variable  which 
contains  a  periodic  discontinuity. 

Both  localization  and  delocalization  depend  on  a  spoiling  of  the  strict 
periodicity  of  a  system.  There  are  many  ways  to  effect  this  "spoilage":  one 
which  will  be  used  extensively  here  and  which  will  be  used  to  establish  a 
benchmark  for  quantification  of  the  phenomena,  is  to  randomize  the  lengths  of 
the  connected  ID  systems.  The  prototypical  complex  of  connected  ID  systems 
is  shown  in  Fig.  1  and  it  is  the  length  (b)  of  the  constituent  systems, 
enumerated  1-N,  which  are  to  be  randomized,  i.e. 


1702 


=  b(l  +  aRn)  ,  0<a<  1  , 

where  Rn  is  a  random  number  between  -1  and  1,  and  a  controls  the  degree  of 
randomization:  e.g.,  a  =  0.1  is  referred  to  as  10%  randomization  and  the 
lengths  are  uniformly  distributed,  0.9  <  (ln)<  1-R  The  benchmark  from 
which  to  quantify  localization  and  delocalization  which  is  proposed  here  is  the 
limit  which  is  approached  when  a  system  is  completely  randomized. 
"Complete  randomization"  as  used  here  means  that  a  =  1  and  if  an  ensemble 
average  is  taken  over  a  large  number  of  realizations  of  Fig.  1,  each  with  a 
different  set  of  Rn’s  then  the  oscillatory  nature  of  the  phase  change  m  the 
amplitude  of  a  wave  traversing  a  system  will  average  to  zero.  With  the  phase 
effects  averaged,  the  diminution  of  a  wave  traveling  through  the  set  of  systems 
will  depend  on  the  average  propagation  loss  factor, 

"■il,  ”■  ' 

and  the  cumulative  transmission  coefficient, 

T^(co)  =  n  T„(cb)  .  (ic) 

Without  the  ensemble  averaging  over  realizations  of  Fig.  1,  each  with  a 
different  set  of  Rn,  the  phase  effects  must  be  accounted  for.  This  may  be  done 
through  the  use  of  an  impulse  response  matrix 

g(x  I  x',  Q))  =  (gjjCXj  I  x'i ,  (»))  ,  (2a) 

in  the  sense  that  it  relates  a  response  in  system]  at  the  coordinate  location  xj  to 
an  impulse  in  system  i  at  the  coordinate  location  x'j  .  More  generally,  in  linear 
systems  in  which  wave  solutions  superpose,  Eq.  (2a)  is  the  Green  function 
kernel  which  relates  the  response  at  any  point  in  any  system, 
v(x  ,  co)  =  { Vj(Xj,  co) }  to  a  distributed  drive  at  a  particular  frequency  through 
the  usual  relation; 


1703 


The  impulse  response  matrix,  Eq.  (2a)  is  formulated  for  a  connect  set  of  1-0 
systems  such  as  shown  in  Fig.  1  by  following  waves  as  they  emanate  from 
the  drive  position  x'j  and  propagate  with  multiple  reflections  and  transmissions 

at  the  junctions  between  systems/^) 

Periodic  Complex  -  Pass  and  Stop  Bands 

Equation  (2)  is  used  to  calculate  a  response  and  to  demonstrate  the 
phenomena  of  localization  and  delocalization  in  a  set  of  dynamic  systems  such 
as  shown  in  Fig.  1.  Certain  parameters  must  be  specified  in  order  to  do  this: 
specifically,  the  number  of  systems,  their  lengths,  their  impedances,  and  their 
manner  of  attachment  to  each  other.  Accordingly,  seven  systems  are  assumed, 
all  stationary  in  time  and  all  supporting  a  single,  non-dispersive  wave  type 
which  propagates  according  to: 

V,  (Xj,  CO)  =  V  (x'j,  CO)  ,  (3) 

The  time  dependent  factor,  e‘“^  is  implicit  in  Eq.  (3)  and 

kj  =koj(l-irij) 

koj  =  co/Cj 

Cj  =  the  phase  speed  of  waves  propagating  in  the  j^h 
system,  generally  may  depend  on  frequency 
(i.e.  dispersive)  but  will  be  taken  to  be  constant 
here. 

■q.  =r  lO”^,  the  propagation  loss  factor  for  the  j^h 
system 

=  the  length  of  the  jtk  system,  bis  the  unperturbed 

length 

mj  =  the  mass  of  the  bead. 


The  response,  as  calculated  by  Eq.  2  and  Reference  5  for  an  infinite  flexible 
string  with  8  equally  spaced  point  masses  (beads)  attached  is  shown  in 
Fig.  2a.  For  the  periodic  complex  all  lengths  are  the  same  ^  j  =  b;  also  cj  =  c, 
rij  =  Ti  and  mj  =  m.  The  complex  is  driven  in  the  first  bay  as  indicated  and 
the  magnitude  of  the  ratio  of  the  response  relative  to  the  response  at  the  drive 
point  x'  is  plotted  as  a  function  of  position  in  the  complex.  The  drive 
frequency  is  (co/co^)  =  10  where  cOj  is  the  fundamental  resonance  of  the 
unperturbed  bay,  i.e.  cOj  =  7tb/c..  The  beads  are  concentrated  masses  equal  to 
0.3  times  the  mass  of  the  string  in  the  unperturbed  bay.  The  standing  wave 
patterns  in  the  bays  are  evident  as  is  the  exponential  decrement  of  the  response 

as  the  excitation  traverses  the  complex. 

Figure  2a  illustrates  the  exponential  attenuation  characteristic  of  wave 
propagation  through  a  cascade  of  connected  one-dimensional  systems.  The 
attenuation  shown  here  is  due  to  the  combined  effect  of  the  propagation  loss 
factors  of  the  individual  bays  (this  is  negligible  here)  and  the  phase  interference 
due  to  the  reverberation  within  the  bays.  The  degree  of  attenuation,  i.e.  the 
slope  of  the  log-response  vs.  distance  curve  is  a  function  of  frequency,  and  it 
is  this  variation  with  frequency  which  gives  rise  to  pass/stop  band  phenomena. 
Accordingly,  a  series  of  responses  of  the  type  shown  in  Fig.  (2a)  are 
calculated  for  a  set  of  increasing  frequencies  and  shown  in  Fig.  (2b).  The 
frequency  range  encompassed  by  the  calculation  is  0  <  (0)/c0|)  ^  20  and  the 
pass  and  stop  bands  appear  as  frequencies  where  the  complex  is  transparent  or 
opaque  respectively  to  the  waves  traversing  the  complex. 

An  alternate  presentation  of  the  response  which  shows  the  pass/stop  band 
phenomena  more  explicitly,  is  the  response  at  the  end  of  the  complex, 
essentially  the  transmissibility  of  the  complex,  as  a  function  of  frequency. 
The  plot  in  this  format  which  corresponds  to  the  parameters  of  Fig.  (2b)  is 
shown  in  Fig.  (2c).  The  peaks  in  the  response  shown  here  correspond  to  the 
pass  bands  of  the  complex  and  are  identical  to  the  pass  bands  as  identified  in 
Fig.  (2b)  as  frequencies  where  the  response  shows  minimal  attenuation  in  x. 
The  minima  in  Fig.  (2c)  are  the  stop  band  frequencies  which  are  more  difficult 
to  identify  in  Fig.  (2b)  because,  for  clarity,  the  data  shown  there  is 
"shadowed"  by  curves  in  the  foreground.  The  benchmark  attenuation  for  the 


1705 


system  discussed  earlier  which  delineates  between  pass  and  stop  band 
behavior  is  shown  by  the  dashed  line  in  Fig.  (2c) 

A-Periodic  Complex  -  Localization  and  Delocalization 

The  phenomena  of  localization  and  delocalization  are  demonstrated  by 
calculating  the  response  of  the  complex  where  the  lengths  of  the  individual 
systems  are  randomized.  Accordingly,  a  system  response  with  all  parameters 
the  same  as  in  Fig.  (2)  except  that  a  =  0.2  in  Eq.  (la)  is  shown  in  Fig.  (3). 
Note  the  dismption  of  regularity  in  the  decrement  as  the  excitation  traverses  the 
systems.  The  transmissibility  shown  in  Fig.  (2c)  for  the  regular  complex  is 
repeated  for  the  randomized  one  and  shown  in  Fig.  (3c).  It  should  be  kept  in 
mind  that  the  calculations  shown  in  Fig.  (3)  are  for  a  specific  realization  of 
randomized  lengths  and  would  not  be  exactly  the  same  for  a  different 
realization  of  the  random  number  set  used  to  generate  the  lengths;  different 
realizations,  however,  show  similar  behavior.  The  dashed  line  in  Fig.  (3c), 
again  represents  the  benchmark  attenuation. 

The  demonstration  of  localization/delocalization  lies  in  the  fact  that  the 
peaks  and  valleys,  respectively,  of  the  transmissibility  shown  in  Fig.  (3c) 
have  moved  toward  the  benchmark  value.  This  is  more  true  at  higher 
frequencies  where  the  length  perturbations  represent  a  larger  fraction  of 
wavelength. 


Special  Cases 

Two  more  complex  structures  are  considered.  The  first  is  an  infinite 
membrane  with  an  array  of  8  stiffeners  or  ribs  attached  as  shown  in  Fig.  4a. 
In  the  unperturbed  case,  the  rib  spacings  are  all  equal  and  of  length  b  and  the 
ribs  are  modeled  as  line  impedances.  The  structure  is  driven  and  the  response 
is  assessed  along  lines  parallel  to  the  ribs  at  positions  x'  and  x  respectively,  as 
shown.  It  should  be  noted  that  the  pass/stop  band  and  localization  phenomena 
do  not  depend  strongly  on  the  magnitude  or  nature  of  the  rib  impedances;  i.e. 
they  can  be  mass-like,  resistive,  spring-like,  resonant,  etc.  The  pass-  and 
stop-band  characteristics  for  this  structure  where  the  rib  impedances  are  mass¬ 
like  with  a  mass  equal  to  0.3  times  the  mass  in  a  bay  of  length  b  are  shown  in 
Fig.  4b.  The  solid  curve  is  the  transmissibility  vs.  frequency  of  the  periodic 


1706 


(unperturbed)  case,  and  the  dashed  curve  is  the  aperiodic  (perturbed)  case  with 
a  =  0.2.  Again,  the  normalizing  frequency  co,  is  the  fundamental  resonance 
of  the  system  of  length  b.  It  should  also  be  noted  that  if  a  thin  plate  were  used 
in  place  of  a  membrane,  the  data  shown  in  Fig.  4b  would  be  similar  but  show 
a  progressive  widening  of  the  frequency  separation  of  adjacent  bands  with 
increasing  frequency  due  to  the  dispersive  nature  of  the  plate  free-wave 
propagation.  The  membrane  was  given  a  propagation  loss  factor  of 
q  =  5  X  10"^  in  the  direction  of  propagation  and  the  slight  progressive 
decrement  of  the  pass-band  peaks  with  frequency  reflects  this;  i.e.  the  structure 
would  exhibit  unity  transmission  at  the  center  of  the  pass  bands  if  there  were 

no  loss. 

A  second  example  of  locaiization/delocalization  effects  in  more  complex 
systems  is  the  planar  array  of  connected  one-dimensional  systems  shown  in 
Fig.  5a.  The  array  dimensions  are  i3b  x  7b  where  b  is  the  length  of  the 
unpermrbed  system,  and,  for  this  example,  the  bead  impedances  were  taken  to 
be  zero.  The  drive  and  assessment  positions  were  at  x  ,  =  0.3b  and 
=  0.5b  respectively,  as  shown.  The  transmissibility  of  the  regular 
(unperturbed)  complex  is  shown  as  a  function  of  (co/co j)  by  the  solid  curve  in 
Fig.  5b  and  one  realization  of  a  perturbed  case  with  a  =  0.2  by  the  dashed 

curve. 


Conclusions 

The  phenomena  of  localization  and  delocalization  is  demonstrated  in  several 
mechanical  systems.  The  canonical  system  is  an  infinite  string  with  a  finite 
array  of  beads  attached  in  which  wave  localization  and  delocalization  occur  in 
the  pass  and  stop  frequency  bands  respectively  when  the  strict  penodicity  of 
the  bead  spacing  is  disturbed.  The  phenomena  are  also  demonstrated  in  a 
membrane  with  a  finite  array  of  attached  ribs.  The  final  example  is  a  two 
dimensional,  planar,  array  of  beads  connected  with  wave-bearing  elements. 
Strong  localization  and  delocalization  occur  in  this  system  also. 


1707 


References 


[1]  Hodges,  C.  H.,  1982.  Journal  of  Sound  and  Vibration  S2,  411-424. 
Confinement  of  vibration  by  structural  irregularity. 

[2]  G.  Maidanik  and  J.  Dickey  1991  Acoustica  73,  119-128.  Localization 
and  delocalization  on  periodic  one-dimensional  dynamic  systems. 

[3]  Langley,  R.  S.,  1995,  Journal  of  Sound  and  Vibration,  188  (5),  717- 
743,  Wave  transmission  through  one-dimensionai  near  periodic 
structures:  optimum  and  random  disorder. 

[4]  D.M.  Photiadis,  1996,  Applied  Meek.  Reviews  49  (2),  100-125,  Fluid 
loaded  structures  with  one-dimensional  disorder. 

[5]  Anderson,  P.W.,  1958,  Physical  Review  109,  1492-1505.  Absence  of 
diffusion  in  certain  random  lattices. 

[6]  L.  J.  Maga  and  G.  Maidanik,  1983,  Journal  of  Sound  and  Vibration,  88, 
473-488.  Response  of  multiple  coupled  dynamic  systems. 


0 


System  #  1, 2,  etc. 


•  •  •  • 

Beads 


Fig.  1.  The  prototypical  complex  of  coonected  one  dimensional  systems,  an  infinite 
string  with  seven  eaually  spa^  beads  mtached. 


1708 


c) 


Fig.  2.  Periodic  bead  spaceing;  a)  the  positions  of  the  beads  and  the  response  as  a 
function  of  distance  along  the  string  at  a  single  frequency,  b)  spatial  responses  as 
in  a)  but  at  a  series  of  frequencies,  and  c)  the  response  at  bead  number  7  as  a 
function  of  frequency,  the  dashed  line  is  the  product  sum  of  the  transmission  at 
beads  1  to  7  inclusive  and  deleniates  between  pass  and  stop  bands. 


1709 


c) 

Fig.  3.  A-periodic  bead  spaceing;  in  the  same  format  as  Rg.  2. 


1710 


b) 

Rg.  4  a)  An  infinite  membrane  with  14  ribs  attached,  b)  The  tMsmissibility 
as  a  function  of  frequency  for  the  regularly  spaced  ribs  (solid  line)  and  for 
one  realization  of  a  set  with  20%  random  variation  in  rib  spaceing  (dashed 
line). 


Rg.  5  a)  A  13  X  7  planar  array  of  connected  1-D  systems  driven  at  x'  in  system 
#1 ;  b)  the  transmissibility  as  measured  in  system  #202  as  a  function  of  frequency 
for  the  regular  system  (solid  line)  and  for  one  realization  of  systems  with  20% 
random  variation  in  lengths  (dashed  line). 


1711 


Estimated  Mass  and  Stiffness  Matrices  of  Shear  Building  from 

Modal  Test  Data 


Ping  Yuan,  ZhiFeng  Wu,  and  XingRui  Ma 
Department  of  Civil  Engineering, 

Harbin  Institute  of  Technology,  Harbin,  150001,  P.R.China 


ABSTRACT: 

A  method  to  estimate  mass  and  stiffiiess  matrices  of  shear  building  from 
modal  test  data  is  presented  in  this  paper.  The  method  depends  on  that 
measurable  points  are  less  than  the  total  structural  degrees  of  freedom,  and  on 
first  two  orders  of  mode  of  structure  are  measured.  So  it  is  applicable  to  most 
general  test.  By  giving  a  method  to  estimate  modal  data  of  immeasurable  points, 
global  mass  and  stiffness  matrices  of  structure  are  obtained  by  using  first  two 
orders  of  modal  data.  By  use  of  iteration  the  optimum  global  mass  and  stiffhess 
matrices  are  gained.  Finally  an  example  is  studied  in  this  paper.  Its  result  shows 
that  this  method  is  reliable. 

l.Introduction: 

Measurement  of  the  dynamic  characteristics,  natural  frequencies  and  mode 
shapes  of  structure  from  modal  dates  is  developed  in  recent  years.  Naturally, 
there  is  only  a  finite  number  of  points  on  the  structure  for  which  data  can  be 
collected.  These  points  are  generally  a  small  subset  of  the  total  degrees  of 
freedom(DOF)  in  a  finite  element  model  of  the  structure.  In  fact,  the  number  of 
measurement  points  may  also  be  less  than  the  total  number  of  vibrational 
modes  identified  in  the  test,  specially  when  utilizing  modem  instrumentation 
with  high  sampling  rates  and  powerful,  inexpensive  scientific  workstations  for 
data  for  n  mode,  where  n>f  there  is  not  a  unique  model  of  classical  mass  and 
stiffhess  form  with  physical  DOF  that  possesses  order-n  dynamics  given  only  1 
spatial  measurement  points. 

Much  research  in  recent  years  has  focused  on  methods  for  correlation  or 
reconciliation  of  finite  element  models  that  inherently  possess  very  large-order 
dynamics  to  the  limited  sensor  and  frequency  data  obtained  from  modal 
testing. 

The  primary  goal  of  the  present  paper  is  to  investigate  direct  solutions  to  the 
inverse  vibration  problem  when  the  number  of  sensors  /  is  less  than  the  number 
of  identified  modes  n.  We  will  show  that  mass  and  stiffhess  matrices  of 


1713 


dimension  n,  referring  to  the  total  DOF,  can  be  found  from  the  I  identified 
modes  and  the  modal  dates  of  the  I  points. 

In  this  paper,  the  unmeasured  modal  dates  are  evaluated  by  the  measured 
modal  dates.  Finally,  the  mass  and  stiffhess  matrices  are  obtain  by  first  two 
order  modes. 

Z.Evaluation  of  unmeasured  modal  dates 

Recent  work  in  the  area  of  structural  identification  has  included  the 
determination  of  mass  and  stiffiiess  matrices  directly  from  continuous  time 
system  realizations.  ^  This  approach  is  that  it  requires  the  dimension  of  the 
physical  mode  to  be  equivalent  to  the  number  of  second-order  states,  implying 
that  the  number  of  independent  sensors  measured  are  equal  to  the  number  of 
identified  modes.  A  more  practical  approach  is  to  enrich  the  computer  mass 
and  stiffness  matrices  with  the  complete  set  of  measured  modes,  independent  of 
the  number  of  sensors.  This  allows  the  resulting  to  express  contributions  of  all 
of  the  modes  observable  from  the  available  sensors.  We  begin  by  developing 
the  concept  of  reduced  the  measured  modal  dates. 

If  structure  is  regard  as  shear  building,  we  can  assume  that  1)  the  total  mass 
of  the  structure  is  concentrated  at  the  levels  of  the  floors;  2)  the  floors  are 
infinitely  rigid  as  compared  to  the  columns;  and  (3)  the  deformation  of  the 
structure  is  independent  of  the  axial  forces  present  in  the  columns.  These 
assumptions  transform  the  problem  from  a  structure  with  an  infinite  number  of 
degree  of  freedom  to  a  structure  that  has  only  as  many  degrees  as  floor  levels. 
Clearly  the  stiffiiess  matrix  of  shear  building  is  a  triangular  matrix,  and  the  mass 
matrix  is  a  diagalization  matrix.  The  mass  and  stiffhess  matrices  of  shear 
building  can  be  written  as: 

^,+^z  -K  .  ® 

-k,  fcj+fc,  -fc,  : 

K=  •••  (1) 

-K-,  K-,+K  -K 

0  .  -K  K  _ 

~m^  •••  0 

M=  :  **.  : 

_0  - 

High  building  always  has  standard  floor  levels.  It  is  to  said  that  these  levels 
have  the  same  mass  and  stiffness.  Assumed  that  the  standard  levels  are  i. 


1714 


i+1 . .j,  j<n,  where  n  is  the  total  levels  of  building.  In  a  word  there  are  j- 

i+l=d  standard  levels,  and  the  frist  level  is  not  standard  level.  If  i-1  and  j  levels 
are  placed  sensors,  these  two  points  become  measured  points.  We  will  use  the 
modal  dates  of  these  two  points  to  evaluate  the  modal  dates  of  unmeasured 
points. 

Suppose  that  we  are  given  an  arbitrary  undamped  MDOF  system.  The 
differential  equation  of  the  system  is  given  by  : 

MX+KX=F 

where  M  is  the  structural  mass  matrix,  K  is  the  structural  stiffness  matrix.  X  is 

the  vector  of  generalized  modal  deflections,  X  is  the  vector  of  generalized 
modal  accelerations.  F  is  the  vector  of  external  forces.  Considered  the 
corresponding  characteristic  equation: 

where  is  the  /th  eigenvalue,  and  (Ji^is  the  /th  corresponding  mode  shape  or 
eigenvector.  Eq(l)  can  be  also  written  as: 

(\kaa  0  1  rm„„  0  0 

K  K.  K.  0  rn,,  0  k  =0  (2) 

[[o  K,  /c„J  [o  0 

where  a-l,2, . i-1,  b=i,i+T . j-C  c=j,j+l, . n,  the  mass  matrices 

m. .  m  are  diaaanalization  matrices,  and  the  stiffness  matrices  can  be  written 

00  ^  CC  w 

as: 

"0  -  -k- 

0  0  j 

■  0  •••  0~ 

and/c^,  =  :  :  =kl  Q) 

... 

Eq.(2)  is  partition  of  Eq.(l).  The  second  line  of  Eq.(2)  can  be  expressed  as: 

^ba^a  '^^bb^b  ^oc  ~^l^bb^b 

This  reduction  concept  considers  the  influence  of  mass.  Substituting  Eq.(3) 
into  Eq(4),  we  have: 

0 

Kb<^b  -  ^i^bb^b  =  '  f 

0 


1715 


Expanding  Eq.(5),  we  have: 

(fc,  +fci,,)<f),- 

+(fc,>,  +fc,V2 )<!),>,  =  0 


.  (6) 

)<!>;-,  =  <<J^J 

Because  from  level  /  to  level  j  are  standard  level,  so  their  stiffness  and  mass 
have  following  relationships. 


m  =  m^-  =•••=  rrij 

:ituting  Eq.(7)  into  Eq.(6) 

rr 


aK}  = 


2-Xia  -1 

1  2-X,a  -1 


0 


0  •••  -1  2-X^aj) 

where  a  =  j,  {<t>6}  =  {(!>,, <!);-, using  least 
square  approach  the  unmeasured  modal  shape  }  can  be  obtained: 


< 

>  =2  < 

0 

> 

'-K 

-nO,- 

-A, 

l  =  D  (8) 


{(}),}  =  (A^rM'^'D  (9) 

By  mean  of  Eq(9)  those  unmeasured  DOF  can  be  expressed  by  measured 
DOF.  It  is  to  said  those  modal  dates  from  i  to  j-1  can  be  evaluated  by  (i-l)th 
and  jth  level  modal  dates.  In  this  method  the  influence  between  the  mass  of 
DOF  and  level  displacement  is  considered.  Using  this  method  measured  points 
can  be  reduced. 


3.The  evaluation  of  mass  and  stiffness  matrices 


Expanding  Eq(l)  we  have: 

(kj+k^^i,  -  Mis  =0 

+(fc2  =0 

.  (10) 

-K-,K-2  +<-K-,  -fc.'t'to  =0 

In  order  to  evaluate  mass  and  stif&ess  of  structure,  equation  (10)  can  be 
written  as: 


1716 


(11) 


-K^rl  4>r;-^r2 

m, 

^12  ^'^12  ‘1^/2"' ^13 

^2 

4>r2*‘J’r;  V^r2  ^r2  “ ‘l>r3 

m, 

■  1 

'l>ln-7“'l^ln-2  ‘1^ In-;  ~  *1*11, 

: 

: 

K 

<j^r75 

where  r  is  the  rth  mo(ie.  This  equation  can  be  simply  wntten  as: 

B{b}  =  0  (12) 

where  {b}  =  . .  In  order  to  solve  this  equation  the  number 

of  equations  should  equal  the  number  of  unknown.  Because  the  number  of 
equation  is  2Xn,  so  we  used  two  order  modes.  Clearly  matrix  B  is  a  singular 
matrix.  In  order  to  assuming  rn^  =  \,  the  Eq.(12)  can  be  expressed  as: 

f  0  1 


(13) 


where  B’  is  the  matrix  that  eliminate  the  last  two  lines  and  last  two  columns  of 
matrix  B.  {b’}  is  the  vector  that  eliminate  the  last  member  of  vector  {b}. 
Using  least  square  approach  (b’}  can  be  obtained. 

f  0  1 


Mir, 


(14) 


The  mass  and  stiffness  matrices  obtained  by  Eq.(14)  are  not  the  real  value. 
There  is  a  constant  rate  between  evaluate  value  and  real  value. 

The  following  procedure  has  developed  for  evaluated  the  global  mass  and 
stiffness  matrices  of  shear  building. 

Step  1:  Choose  the  initial  parameter 

a  is  the  ratio  between  the  mass  and  the  stifhiess  of  standard  levels.  We 


1 

defined  the  initial  parameter  =  —  . 

Step  2:  Evaluate  the  mode  shapes  of  unmeasured  DOF 

In  this  step  Eq.(9)  is  used  to  estimated  the  two  order  mode  shapes  and 

.  Where  and  |(j)}  are  the  /th  and  rth  corresponding  mode  shapes. 


1717 


Table  2. 


ki(N/m) 

k2 

ks 

k4 

mi(kg) 

m2 

nu 

Real  value 

18000X 

6 

18000 

X3 

18000 

X5 

18000 

X2 

18000 

X2 

18000 

X2 

Evaluation 

2.9948 

0.99896 

0.99896 

1.0 

5.  Conclusion 

A  new  method  to  evaluated  mass  and  stiffness  matrices  of  shear  building 
from  modal  test  dates  is  presented  in  this  paper.  This  approach  based  on  the 
number  of  measurement  points  less  than  the  number  of  total  degrees  of 
freedom  of  structure.  In  this  method  only  first  two  order  modes  are  used  and 
the  influence  of  mass  is  considered.  By  mean  of  this  method  it  is  possible  of 
using  modal  test  dates  to  evaluate  the  global  mass  and  stiffhess  matrices  and 
the  damage  of  structure. 

6.  References 

1.  Kabe,A.M.  Stiffenss  Matrix  Adjustment  Using  Mode  Data.  AIAA.  1985. 
Vol23.  No.9.  1431-1436 

2.  Farhat,C.,  and  Hemez,  F.,  Updating  Finite  Element  Dynamic  Models  Using 
an  Element-By-Element  Sensitivity  Methodology.  AIAA,  1 993, Vol  31, 
No.9. 1702-1711 

3.  Yang,C.D.,  and  Yeh,F.B.,  Identification,  Reduction,  and  Retinement  of 
model  Parameters  by  the  Eigensystem  Realization  Algorithm.  J.  of  Guidance, 
Control  and  Dynamics,  Vol.  13, No. 6,  1990,  1051-1059 

4.  Baruch, M.,  Optimization  Procedure  to  Correct  Stiffness  and  Flexibility 
Matrices  Using  Vibration  Tests,  AIAA,  Vol.  16,  11,  1978,  1209-1210 

5.  Topole,K.G.,  and  Stubbs,N.,  Non-destructive  Damage  Evaluation  of  a 
Structure  from  Limiten  Modal  Parameters,  Earthquake  eng.  struct,  dyn.,  Vol 
24,  11,  1995,  1427-1436 


1718 


THE  PROBLEM  OF  EXPANDING  THE  VIBRATION  FIELD  FROM 
THE  MEASUREMENT  SURFACE  TO  THE  BODY  OF  AN  ELASTIC 

STRUCTURE 


Yu.LBobrovnitskii 

Blagonravov  Institute  of  Mechanical  Engineering  Research  of  the  Russian 
Academy  of  Sciences,  Moscow  101830,  Russia 


Introduction 

Knowledge  of  the  distribution  and  magnitude  of  the  dynamic  stresses  of 
an  engineering  structure  due  to  extensive  vibration  is  important  for  the 
estimation  of  the  structure  reliability  and  its  mean  service  time.  Another 
practical  problem  where  it  is  needed  is  control  of  the  structural  vibration: 
knowledge  of  the  stresses  and  displacements  allows  one  to  compute  the  vector 
field  of  the  vibration  power  flow  which  makes  visible  the  sources  and 
transmission  paths  of  vibration  and  thus  indicates  adequate  means  to  control  it 
[1,2]. 

Commonly  used  sensors,  e.g.,  accelerometers  or  strain  gauges,  can 
measure  the  vibration  parameters  (acceleration  or  strains)  in  discrete  points. 
Some  instruments  developed  in  the  last  decade  such  as  laser  interferometers  [3] 
and  the  vibrometers  based  on  the  near  field  acoustic  holography  [4],  can 
measure  continuous  distribution  of  the  vibration  amplitudes  at  a  certain  surface. 
Nevertheless  in  practice,  at  best  only  a  part  of  the  structure  surface  is  accessible 
for  measuring  vibration  directly.  For  most  of  the  interior  points  of  the  structure 
body  as  well  as  for  some  portions  of  its  surface,  mounting  sensors  is  impossible 
or  impractical.  So,  the  only  way  of  estimating  the  stress-strained  state  of  the 
whole  structure  is  to  expand  the  vibration  field  from  the  measurement  surface 
to  the  structure  body. 

In  this  paper,  an  approach  to  such  expansion  is  proposed  which  is 
applicable  to  complex  elastic  structures  of  which  a  part  of  the  surface  is 
accessible  for  direct  vibration  measurements  while  the  rest  of  the  surface  and 
the  body  are  not.  This  approach  consists  of  measuring  the  distribution  of  the 
amplitudes  for  three  components  of  the  vibration  displacement  (or  acceleration) 
on  a  portion  of  the  accessible  surface.  For  the  volume  contiguous  with  this 
portion,  a  special  boundary-value  problem  which  is  called  here  the  problem  of 
field  reconstruction  (FR-problem)  is  stated  and  solved  (see  Fig.l).  As  a  result, 
the  displacement  (and  stress)  field  of  this  volume  is  determined  (reconstructed) 
through  the  measurement  data.  Then,  using  the  vibration  amplitudes  measured 


1719 


on  another  portion  of  the 
accessible  surface,  one  can 
similarly  determine  the 
displacements  and  stresses  in 
another  volume,  etc.,  until 
the  whole  structure  is 
investigated. 

The  idea  of 
reconstructing  the  general 
pattern  from  incomplete  or 
indirect  data  is  not  new  and 
is  often  used  in  mechanics 
and  structural  dynamics.  It  is 
used,  for  example,  in  the 
mode  shape  expansion 
methods  of  the  model 
updating  techniques,  where 
the  responses  measured  on  a 
part  of  the  DOF’s  of  the  FE- 
model  are  expanded  to  the  slave  (unmeasured)  DOF’s  using  the  equations  of 
motion  [5],  Another  example  is  reconstruction  of  the  time  history  and 
localisation  of  external  forces  from  the  measured  structural  responses  [6]. 

All  the  cases,  where  the  idea  of  reconstruction  is  realised,  are  distinguished  by 
the  statement  of  the  corresponding  mathematical  problem  and  by  physical 
peculiarities.  In  this  paper,  the  key  problem  is  stated  as  reconstruction  of  the 
vibration  field  in  a  finite  elastic  solid  from  the  amplitudes  of  the  displacements 
of  a  part  of  its  boundary  free  of  tension.  In  structural  dynamics,  such  statement 
was  first  proposed  in  Ref  [7]  relating  to  measurement  of  the  vibration  power 
flow  in  solids.  Similar  statements  are  met  in  a  number  of  papers  on  the  static 
theory  of  elasticity  (e.g.,  [8])  for  reconstruction  of  the  stresses  inside  a  machine 
part  through  the  measurement  of  the  strains  on  its  surface. 

In  what  follows,  the  rigorous  mathematical  formulation  of  the  FR- 
problem  is  given  and  its  general  properties  are  studied.  Results  of  the  computer 
simulation  and  of  the  laboratory  experiment  are  presented  which  prove  the 
practicality  of  the  approach  suggested. 


u(s)=uo  f(s)=0  (b) 


Fig.  l.An  elastic  structure  (a)  of  which  only  the 
upper  surface  S  is  accessible  for  vibration 
measurement;  (b)  boundary  value  problem  for 
the  chosen  volume  V:  displacements  u  and 
force  f  are  specified  on  S. 


Formulation  of  the  problem 

For  a  selected  elastic  volume  V  of  the  structure  (Fig.  lb),  the  problem  of 
the  harmonic  field  reconstruction  can  be  formulated  as  follows  (time 
dependence  exp(-icot)  is  omitted); 
find  a  solution  to  the  homogeneous  Lame  equations 


1720 


(1) 


^JhM(x}+{X+|ijgxz.&.Av^u(x)+p(^u(x)=■0,  x&V, 

which  satisfies  the  conditions 

u(s)  =  uo(s),  f(s)  =0,  s  eS,  (2) 

on  the  upper  part  S  of  the  surface.  Here  x=  (xj,  X2,  X3)  are  the  co-ordinates  of  a 
point  of  the  volume  V,  u  =  (ui,  U2,  U3)  is  the  displacement  vector,  f=(f},f2, 
fs)  is  the  vector  density  of  the  forces  acting  on  the  surface,  X  and  /i  are  the 
elastic  Lame  coefficients,  and  Uo(s)  are  the  known  (measured)  displacement 
amplitudes. 

The  formulation  (1),(2)  is  not  traditional  for  equations  of  the  elliptic 
type:  the  boundary  conditions  on  the  part  S  of  the  surface  are  overdetermined, 
i.e.,  both  the  displacements  and  the  forces  are  specified,  whereas  no  quantities 
are  specified  on  the  rest  part  0  of  the  surface. 

The  boundary- value  problem  (l),(2)  may  be  formulated  as  an  integral 
equation.  Let  G(x/q)  be  the  3x3  matrix  of  the  Green's  functions  describing  the 
response  (displacement  vector)  at  a  point  x  e  F  to  a  unit  force  at  a  point  q  e 
0.  Assuming  x  =  s  e  S,  the  problem  (1),(2)  may  be  rewritten  as 

uo(s)  =ffG(s/q)f(q)dQ.  (3) 

This  is  a  set  of  three  Fredholm  equations  of  the  first  kind,  where  the  unknowns 
are  the  reaction  forces  f(q)  acting  on  the  inaccessible  surface  0;  the  matrix 
G  is  assumed  to  be  known.  If  the  forces  f(q)  are  found  from  Eq.(3),  then  the 
displacement  field  of  the  volume  V  can  be  computed  as 

u(x)  =f/G(x/q)f(q)dQ.  (4) 

Q 

The  formulations  (1),(2)  and  (3),(4)  are  convenient  for  investigating  the 
general  properties  of  the  problem.  In  practice,  preferable  is  the  formulation 
based  on  the  expansion  of  the  field  in  the  normal  modes  of  V 

u(x)  =Za„q>Jx),  (5) 

where  the  forms  q)„(x)  are  assumed  to  be  known  and  the  amplitudes  a„  must 
be  determined  from  the  first  boundary  condition  (2)  on  S: 

Uo(s)  =Ea^(Pn(s).  (6) 


1721 


Eq.(6)  represents  an  expansion  of  the  known  function  uo(s)  in  terms  of  the 
functions  q)„(x),  non-orthogonal  on  S,  which  can  be  reduced  in  different  ways 
to  a  set  of  linear  algebraic  equations.  In  practice,  the  number  of  normal  modes 
is  taken  to  be  finite,  and  the  number  of  the  unknowns  a„  in  Eqs.(5),(6)  as  well 
as  the  number  of  the  algebraic  equations  is  finite,  too.  Discretization  of  the 
continuous  operators  in  (l)-(3)  also  leads  to  finite  systems  of  linear  equations. 
Thus,  in  all  the  above  formulations  the  problem  of  the  field  reconstruction 
reduces  to  the  linear  operator  equation  of  the  first  kind 


(7) 

where,  in  theory,  the  functions  u(s)  and  f(q)  are  elements  of  two  functional 
spaces  U  and  F,  and  /4  is  a  linear  operator.  In  practice,  u  and  /  are  vectors 
of  two  Euclidean  spaces,  and  A  is  a  rectangular  matrix  operator.  • 


General  properties  and  formal  solution 

The  problem  of  field  reconstruction  has  the  following  general 
properties.  The  problem  has  a  bounded  solution  if  the  vector-function  Uo(s)  is 
sufficiently  smooth  -see  below.  The  solution,  if  it  exists,  is  unique.  It  means  that 
the  absence  of  information  on  the  inaccessible  part  O  of  the  boundary  is 
completely  compensated  by  the  overdetermined  boundary  conditions  (2)  on  the 
accessible  part  .S'.  The  proof  of  the  uniqueness  is  based  on  the  Almansi's 
theorem[9]  according  to  which  if  on  a  part  (even  very  small)  of  the  surface  of  a 
finite  elastic  body  displacements  and  stresses  are  simultaneously  equal  to  zero, 
then  the  stresses  are  zero  everywhere  and  the  body  is  at  rest.  Almansi  proved 
the  theorem  for  statics;  an  extension  to  the  dynamic  theory  of  elasticity  is  given 
in  the  paper  [7].  One  more  property;  the  solution  of  the  FR-problem  does  not 
continuously  depend  on  the  input  data:  small  variations  A«o  of  the  prescribed 
function  may  cause  large  variations  A/  of  the  solution.  This  can  be  concluded 
from  the  general  properties  of  the  Fredholm  integral  equations  of  the  first  kind 
with  continuous  kernels  (the  case  of  Eq.3).  Hence,  the  problem  of  field 
reconstruction  belongs  to  the  class  of  ill-posed  in  the  sense  of  Hadamard 
problems  of  mathematical  physics. 

Ill-posed  problems  are  often  met  and  solved  in  various  scientific 
disciplines.  For  example,  the  mathematical  problem  of  differentiating  functions 
is  ill-posed;  the  mechanical  problem  of  reconstructing  the  external  forces 
through  the  structural  response  mentioned  above  is  also  ill-posed,  etc.[10,ll]. 
All  the  existing  methods  of  treating  such  problems  are  principally  based  on  the 
idea  of  replacing  the  ill-posed  problem  with  a  well-posed  problem  appropriately 
chosen  with  the  aid  of  an  additional  information  concerning  the  desired 
solution.  For  treating  the  field  reconstruction  problem,  the  most  appropriate  is 


1722 


the  Singular  Value  Decomposition  (SVD)  technique[12].  Below  follows  the 
solution  of  the  FR-problem  obtained  by  this  technique. 

Let  aj  >  02  >...  be  the  singular  values  of  the  operator  A  in  Equation 
(7),  and  {fi,  f2,  ...  }  and  (uj,  U2,  ...  }  be  the  singular  pair,  i.e.  two  sets  of 
.  orthonormal  functions  (Note  that  of  are  the  eigen  values  of  the  Hermitien 
operators  A  A  and  AA*,  while  fj  and  Uj  are  their  eigen  functions). 
Representing  a  given  function  w  as  a  series  in  terms  of  Uj  and  seeking  the 
solution  as  a  series  in  fj ,  one  can  obtain  the  following  exact  formal  solution  to 
the  problem  (7): 

0£3  OO 

f=Z(bj /<Jj)  fj ,  where  u  =  Zbj  Uj .  (8) 

It  is  seen  from  Eq.(8)  that  a  bounded  solution  to  the  problem  exists  (the  series 
converges)  if  the  given  function  w  is  sufficiently  smooth  or,  more  exactly,  if 
the  coefficients  bj  of  its  expansion  (8)  tend  to  zero  more  rapidly  than  the 
singular  values  q;- .  For  real  structures,  the  singular  values  tend  to  zero  very  fast 
(exponentially),  therefore  the  exact  functions  of  distribution  of  the  vibration 
amplitudes  at  the  accessible  part  S  of  the  surface  should  be  and  actually  are 
very  smooth  and  do  not  contain  components  rapidly  oscillating  along  the 
surface,  i.e.,  addenda  in  (8)  with  large  numbers  j.  However,  small  but  finite 
random  errors  in  u,  experimental  or  due  to  rounding  in  a  computer,  have  wide 
spatial  spectra  and  result  in  expansions  (8)  with  non-zero  coefficients  bj  of  large 
numbers  which,  after  enhancing  by  the  small  singular  values,  may  cause  large 
errors  in  the  solution  f,  making  it  unstable. 

The  simplest  way  to  overcome  the  difficulty  is  to  seek  an  approximate 
solution  which  involves  a  finite  number  N  of  terms  in  the  singular  value 
decomposition,  i.e.  to  truncate  the  series’  in  (8).  With  a  judicious  choice  of  N 
this  solution  does  not  differ  too  much  from  the  exact  solution  (8),  because  the 
exact  (not  contaminated  by  noise)  vibration  function  u  does  not  contain 
components  with  high  numbers.  On  the  other  hand,  this  solution  cuts  off  the 
rapidly  oscillating  components  j  >N,  thereby  reducing  the  inaccuracy  due  to 
experimental  or  computer  errors  in  the  input  data.  Truncating  the  series  (8),  we 
restrict  the  solution  by  smooth  functions  and  this  is  the  additional  a  priori 
information  which  makes  the  problem  well-posed.  However,  the  choice  of  the 
best  truncation  number  is  a  difficult  and  unstudied  question,  and  answering 
it  was  one  of  the  objectives  of  the  computer  simulation  and  laboratory 
experiment. 


Results  of  computer  simulation 

In  computer  simulation,  two  simple  structures  were  studied  -  a  finite 
longitudinally  vibrating  rod  and  a  thin  elastic  strip  executing  vibrations  in  its 


1723 


plane(Fig.2).The  rod(Fig.2a) 
is  excited  by  a  harmonic 
force  at  the  left  end.  The 
accessible  for  measurement 
part  is  supposed  to  be  the 
region  S  at  the  right  end 
(data  region),  while  the  rest 
part  0  of  the  rod  is 
considered  as  the 
reconstruction  region.  The 
complex  displacement 
amplitude  u(x)  satisfies  the 
Bernoulli’s  equation  of 
longitudinal  vibrations  [13] 
with  a  complex  Young 
modules.  This  boundary 
value  problem  has  an 
analytical  solution  which  has 
been  used  for  computing  the 
input  data  “measured”  in  the  region  S  and  for  estimating  the  accuracy  of 
reconstruction  in  the  region  O.  In  calculations,  a  modal  model  was  used  with 
the  modes  corresponding  to  the  free  boundary  conditions  at  both  ends: 

{ u„(x)  =  (s„/ if  ^  cos  (7t(n  -i)x/l);  Sj  =1,  Sn=2  for  n  =  2,3,...}. 

The  solution  (5)  has  here  the  form  of  a  finite  sum 

A/ 

u(x)  ^  (9) 

with  unknown  amplitudes  a„ .  Equating  the  representation  (9)  to  M  measured 
values  of  u  a.t  M  points  of  the  region  S  gives  a  set  of  M  linear  algebraic 
equations  with  N  unknowns  that  were  solved  by  the  SVD-technique. 

The  elastic  strip  (Fig.  2b)  of  height  2H  comparable  with  the  elastic 
wavelengths  vibrates  in  its  plane.  It  is  supposed  that  only  upper  surface  y  =H 
of  the  strip  is  accessible  for  measurement.  It  is  required  to  reconstruct  the 
vibration  field  in  the  region  V  with  the  following  dimensions:  /  =3H. 
Vibrations  of  the  strip  satisfy  the  dynamic  equations  of  the  plane  stress  state  and 
the  boundary  conditions  of  absence  of  stresses  at  y  =-H  and  y  H.  As  it  is 
known,  they  are  analogous  to  vibrations  of  an  elastic  layer.  Therefore,  the  field 
in  the  strip  is  represented  as  a  sum  of  N  Lamb’s  normal  modes  [13]: 

A//2 

{U:,(x,y):  Uy(x,y)}  =  Z[a„  {u^'"(y):  Uyn(y)}  +  (10) 

b„  {u,cn  (y):  Uy„-(y)}  tx^(ik„(l-x))], 

where  the  K  are  the  propagation  constants,  i.e.,  the  roots  of  the  Rayleigh- 
Lamb  dispersion  equation,  and  the  expressions  in  brackets  describe  the  mode 


/  - 

0 

y 

H 

- ^  (a) 

/  X 

/  ' 

i  1 

\l  X  \ 

-H 

Fig.2.  Two  simp 
simulation:  a  loi 
and  an  elastic  si 

0 

le  structures  used  in  computer 
ngitudinally  vibrating  rod  (a) 
tip  vibrating  in  its  surface(b). 

1724 


forms.  On  the  accessible  surface  jl/  =H  of  the  strip,  M/2  equidistant  points  are 
chosen  at  which  the  displacement  components,  Ux  and  are  computed 
using  the  exact  solution.  (The  exact  solution  is  taken  as  in  an  infinite  strip 
excited  by  a  unit  y-force  applied  to  the  point  (-4H,  H),  i.e.,  at  the  distance  4H 
from  the  region  of  interest  V).  Equating  the  “measured”  amplitudes  to  (10) 
gives  a  set  of  M  linear  algebraic  equations  with  N  unknown  mode 
amplitudes,  and  h„ ,  which  can  be  found  by  the  SVD-technique. 

The  results  of  computer  simulation  obtained  for  the  two  different 
structures  on  Fig.  2  (the  first  structure  is  one-dimensional  and  the  second  is  two- 
dimensional)  are  similar.  They  are  also  similar  to  those  obtained  for  other 
structures  (the  author  verified  them  for  an  acoustic  waveguide  and  for  a 
circular  cylindrical  shell).  So,  they  are  rather  general. 

The  first  and  practically  most  important  result  is  the  existence  of  an 
optimal  model:  there  is  a  number  No  of  normal  modes  (model -parameters) 
which  renders  minimum  to  the  reconstruction  error.  It  means  that  too 
complicated  and  exact  models  containing  an  excessive  number  of  model 
parameters  N  >  No  as  well  as  rough  models  with  a  small  number  of  the 
parameters  N  <  No,  give  larger  errors  of  field  reconstruction  than  the  optimal 
model  with  No  parameters.  The  best  number  No  depends  on  the  structure 
type,  geometry,  frequency,  etc.,  but  most  of  all  on  the  accuracy  and  amount  of 

Fig.  3  presents  the 
field  reconstruction 
error  A  versus  the 
number  of  the  model 
parameters 
computed  for  the 
strip  on  Fig.2b.  The 
error  A  is  defined 
as  relative  square 
module  deflection  of 
the  reconstructed 
displacement  from 
the  exact  one 
averaged  over  the 
reconstruction 
region  V.  In 
computing  these 
plots,  a  random 
error  of  prescribed 
standard  value  S  is 
added  to  the 

“measured”  data.  Four  curves  in  Fig,3  correspond  to  various  values  of  the 


1725 


standard  error  5.  It  is  seen,  that  each  curve  of  the  field  reconstruction  error  has 
a  pronounced  minimum  beyond  which  the  error  sharply  increases  reaching 
arbitrary  large  values.  This  result  seeming  paradoxical  has  a  clear  physical 
explanation.  The  reason  lies  in  random  errors  of  the  input  data. 

If  there  were  no  input  errors,  i.e.,  if  the  input  data  were  known  mathematically 
exactly,  the  reconstruction  error  would  monotonically  decrease  with  the 
number  of  modal  parameters  N:  the  more  exact  is  the  model  the  better  is  the 
approximation.  When  input  errors  are  introduced,  the  decreasing  of  A(N) 
holds  only  to  a  certain  limit  since  for  large  N  the  errors  in  the  input  data, 
enhanced  by  small  singular  values  (see  equation  8),  may  give  an  arbitrarily 
large  error  in  the  result  of  reconstructing  the  field. 


.  15 


Fig.  4.Amplitude(a)  and phase(b)  ofthe  bardisplacement 
exact  -solid  lines,  reconstructed  -  dotted  lines:  kl=4. 7; 
loss  factor  0. 05;  number  of  modes  used  2. _ 


Thus,  for  each 
value  of  the  input 
error,  there  exists 
an  optimal  number 
of  the  model 
parameters  which 
corresponds  to  the 
minimal  error  in  the 
reconstructed  field. 
Curves,  similar  to 
the  curves  in  Fig.3, 
are  obtained  also 
for  other  structures. 
For  example,  when 
30%  of  the  bar  in 
Fig.2a  are 

accessible  for 
measurement  and 
the  input  error  is 
equal  to  10*^,  the 
best  number  of 
modes  is  No  ""12. 
Fig.4-6  present  the 
displacement 
amplitude  and 
phase  distributions 
along  the  rod  for  a 
rough  model  {N 
=2),  the  optimal 
model  (N=J2)  and 
an  excessively 
complicated  model 


1726 


(N=  30),  It  is  seen 
that  in  all  cases  the 
reconstruction 
error,  being  small  in 
the  measurement 
region  S,  increases 
with  distance  from 

Another  finding 
in  the  computer 
simulation  concerns 
the  amount  of  input 
data  necessary  for 
obtaining  the  best  solution.  Fig.7  presents  a  typical  dependence -of  the  field 
reconstruction  error  on  the  number  M  of  measurement  points  in  the  region  S 
for  alongitudinally  vibrating  bar  (Fig.2a). 

It  is  seen,  that  the  reconstruction  error  is 
unstable  when  the  number  M  is  small. 

With  increasing  M  the  error  A 
becomes  stable  and  decreases  tending  to 
a  certain  constant  value.  Similar 
dependencies  take  place  for  the  strip  and 
other  structures.  The  optimal  amount  of 
the  input  data  correspond  to  the  number 
of  measurements  M  two  or  three  times 
the  number  N  of  the  model  parameters 
involved.  A  further  increase  of  the 
amount  of  data  does  not  improve  the 
results  and  is,  therefore,  unjustified. 

It  should  be  noted  that  all 
components  of  the  prescribed  displacement  vector  are  needed  in  order  to  obtain 
the  best  reconstruction  error.  This  follows  from  the  uniqueness  theorem 
mentioned  above:  the  problem  of  field  reconstruction  is  uniquely  solvable  only 
if  the  full  displacement  vector  on  the  accessible  surface  is  available. 


Some  experimental  results 

To  verify  the  proposed  method,  a  laboratory  experiment  has  been 
carried  out  on  a  circular  cylindrical  shell  (Fig.  8).  A  finite  open  shell  with 
dimensions  in  mm  900x300x3.5  is  excited  by  a  shaker  with  harmonic  signals. 
The  amplitudes  and  phases  of  vibration  are  measured  at  the  surface  5'  by  a 
small  (2g)  3-component  accelerometer,  the  data  are  fed  into  a  PC.  The 


1727 


amplitudes  and  phases  of  vibration  in  the  region  O  are  reconstructed  (by  the 
algorithm  described  above)  and  compared  with  the  actual  ones.  The 
reconstruction  error  is  computed  with  respect  to  the  actual  field  of  the  region 
0.  In  modelling  the  vibration  field  in  the  shell,  the  displacement  vector  is 
represented  by  the  series  of  the  normal  modes  which  satisfy  the  simplest  shell 
equations  (the  Donnel-Mushtari  theory  [14]).  The  maximal  number  of  the 
normal  modes  used  is  52.  These  modes  correspond  to  the  circumferential 
numbers  up  to  m  =10,  The  evanescent  modes  decaying  more  than  40  dB  at  the 
distance  from  the  region  S+Q  to  the  end  of  the  shell  or  to  the  shaker,  are 
excluded  from  the  model.  The  total  error  of  measuring  complex  vibration 
amplitudes  is  estimated  as  0.07  <  S<  0.08. 

Fig.  9  and  10 
show  the  dependencies 
of  the  reconstruction 
error  A  (solid  lines) 
on  the  complexity  of 
models  used  for  one  of 
the  frequencies  (f  = 
700  Hz)  and  for  two 
different  amount  of 
input  data.  The  curves 
are  obtained  as 
follows. 

First,  the  simplest 
models  containing  the 
normal  modes  of  one 
single  circumferential 


1728 


number  m  are  tried. 
Among  1 1  models 
with  m  =  0,1,...,  10, 
the  best  model  is 
chosen  which  gives 
the  minimal  error  in 
describing  the 

measured  data  (in 
practice,  only  these 
data  are  actually 
available).  In  both 
cases  of  Fig.  9, 10  it  is 
the  model  with  the 
circumferential 
number  m  ^  6.  It 
corresponds  to  the 
model  complexity  MC  =  1  in  Fig.  9, 10.  Then,  all  models  with  two 
circumferential  numbers  (MC  =  2  in  Fig.9,10j  are  considered:  one  is  mj  =  6 
and  another  is  chosen  among  10  models  with  m2  ^6  which  gives  the  minimal 
error  in  describing  the  input  data  (It  is  found  that  m2  equals  5  in  the  case  of 
Fig.9  and  m2  ^4  for  Fig.  10).  After  that,  all  models  with  three  circumferential 
numbers  (MC=3)  are  investigated,  etc.,  until  all  the  models  are  exhausted. 
Thus,  the  dashed  curves  in  Fig.9, 10  correspond  to  the  best  models  of  various 
complexity  for  the  measurement  region  S  of  the  shell.  Therefore,  these  curves 
decrease  monotonically,  at  least,  do  not  increase,  with  the  number  of  the  model 
parameters.  At  the  same  time,  the  error  of  the  expansion  to  the  reconstruction 
region  0  (solid  curves  in  Fig.9, 10)  has  tendency  to  increase  with  the  number 
of  the  model  parameters.  More  exactly,  they  have  minima  beyond  which  they 
increase  monotonically.  For  very  complicated  models,  the  reconstruction  error 
may  reach  hundred  of  thousand.  With  these  curves,  correlate  the  curves  of  the 
condition  number  (dotted  lines). 

Now,  the  question  how  to  choose  the  optimal  model  which  minimises 
the  reconstruction  error  in  the  inaccessible  region  (i.e.,  corresponds  to  the 
minimum  of  the  solid  line  curves  in  Fig.9, 10)  using  only  the  information  about 
vibration  of  the  measurement  (accessible)  region  (i.e.,  using  the  dashed  and 
dotted  line  curves  in  Fig.9, 10)  may  be  answered  as  follows.  As  one  can  observe 
from  Fig.9  and  10,  the  error  of  describing  the  measured  data  (dashed  lines) 
first  rather  rapidly  decreases  with  the  model  complexity  and  then  becomes 
almost  constant.  The  best  model  just  corresponds  to  the  transition  from  the 
interval  of  rapid  decrease  to  the  interval  of  stable  values  of  the  dashed  line 
curve.  Conversely,  the  curves  of  the  condition  number  (dotted  lines  in  Fig.  9, 10) 
first  grow  slowly  with  MC  but,  after  the  best  model  is  reached,  their  growth 
becomes  rather  fast. 


1729 


Thus,  it  can  be  concluded  that  the  optimal  model  can  be  approximately 
identified  from  the  behaviour  of  the  curves  of  the  error  in  the  input  data  and  of 
the  condition  number.  From  the  physical  point  of  view,  this  “rule  of  thumb”  is 
obvious  -  see  explanation  of  Fig.3.  But  quantitatively,  it  is  rather  uncertain:  one 
can  only  find  an  interval  of  models  within  which  the  best  model  lies.  E.g.,  for 
the  cases  of  Fig.9,10  the  intervals  correspond  to  MC  ~  {4,5,6}  and  {2,3,4} 
where  the  reconstruction  error  equals  {0.35,  0.25,  1.2}  and  {0.64,  0.75,  0.85} 
while  the  minimal  values  in  these  cases  are  0.25  and  0.51.  Much  more  certain 
identification  of  the  best  model  can  be  made  if  at  least  one  measurement  point 
is  taken  in  the  reconstruction  region;  in  other  words,  if  it  is  possible  to  place  at 
least  one  sensor  into  the  needed  part  of  the  structure.  In  this  case,  however, 
another  procedure  of  the  optimal  model  identification  is  required. 


Summary 

The  problem  of  expanding  the  vibration  field  from  the  measurement 
surface  to  the  volume  of  an  elastic  structure  is  rigorously  posed  and  studied 
theoretically  and  experimentally.  When  formalised,  this  problem,  also  called  as 
the  field  reconstruction  problem,  is  a  non-traditional  boundary  value  problem 
for  differential  equations  of  elliptic  type.  The  conditions  for  existence  of  a 
solution  are  found  and  the  uniqueness  theorem  is  proven.  A  general  solution 
based  on  the  generalised  singular  value  decomposition  is  obtained. 

Results  of  computer  simulation  with  simple  structures  are  presented. 
Most  attention  is  paid  to  studying  the  accuracy  of  the  expansion  to  unmeasured 
parts  of  the  structure.  Relations  between  the  expansion  accuracy,  the 
measurement  errors  and  the  complexity  of  the  vibrational  model  of  the 
structure  (number  of  model  parameters)  are  established.  A  salient  feature  of 
these  relations  is  that,  for  a  given  input  data  accuracy,  there  is  an  optimal  model 
which  minimises  the  expansion  error. 

Results  of  laboratory  experiments  with  a  steel  cylindrical  shell  executing 
forced  harmonic  vibration  are  also  presented  aiming  to  verify  the  obtained 
relations.  Procedures  of  choosing  the  optimal  models  are  discussed. 


REFERENCES 

1.  Ramakumar,  R.,  Reliability  engineering.  Fundamentals  and  applications. 
Prentice  Hall,  Allyn  &  Bacon  and  Ellis  Horwood,1993, 

2.  Structural  intensity  and  vibrational  energy  flow  (Proc.4th  Int.  Congress  on 
intensity  technique).  Senlis,  France,  1993. 

3.  Tyrer,  J.R.,  Determination  of  surface  stresses  and  velocities  by  optical 
measurement.  In  [2],  35-45. 


1730 


4.  Maynard,  J.D.,  Williams,  E.G.  and  Lee,Y.,  Nearfield  acoustic  holography 
(NAH).  JAcoust  Soc.  Am.,  1985,  78,  1395-1413. 

5.  To,W.M.  and  Ewins,  D.L.,  The  role  of  the  generalised  inverse  in  structural 
dyna.mics.  J. Sound  and  Vibr.,  1995,  186,  185-195. 

6.  Yen,  C.S.  and  Wu,  E.,  On  the  inverse  problem  of  rectangular  plates 
subjected  to  elastic  impulse.  J.  Appl  Meek,  1995,  62,  692-705. 

7.  Bobrovnitskii,  Yu.L,  The  problem  of  field  reconstruction  in  structural 
intensimetry;  statement,  properties,  and  numerical  aspects.  Acoust.  Phys., 
1994,  40,331-9. 

8.  Preiss,  A.K.,  Evaluation  of  stress  fields  by  a  finite  body  of  experimental 
'\nformd.i\on.  Mashinovedenie,  1984,  N2,  77-83. 

9.  Almansi,  E.,  Un  teorema  sulle  deformazioni  elastiche  dei  solid!  isotropi,  Atti 
R.Accad.Lincei,  1907,  Ser.5,  16,  865-8. 

10.  Lavrentiev,  M.M.,  Some  improperly  posed  problems  in  mathematical 
physics.  Springer,  Berlin,  1967. 

11.  Tikhonov,  A.N.  and  Arsenin,  V.Ya.,  Solution  of  ill-posed  problems. 
Simon&Schuster,  Washington  DC,  1977. 

12. Golub,  G.H.  and  van  Loan,  C.F.,  Matrix  computations.  North  Oxford 
Academic  Publishing,  Oxford,  England,  1983. 

13.  Graff,  K.F.,  Wave  motion  in  elastic  solids.  Clarendon  Press,  Oxford,  1975. 
H.Leissa,  A.W.,  Vibration  of  shells.  US  Government  Printing  Office, 

Washington  DC,  1973. 


1731 


EVALUATION  OF  THE  EQUIVALENT  GEAR  ERROR  BY 
VIBRATIONS  OF  A  SPUR  GEAR  PAIR 


M.  Amabili  (*)  and  A.  Fregolent  (**) 

(*)  Dipartiraento  di  Meccanica,  Universita  di  Ancona,  Ancona,  Italy 

(**)  Dipartimento  di  Meccanica  ed  Aeronautica,  Universita  di  Roma 
“La  Sapienza”,  Roma,  Italy 


ABSTRACT 

A  new  approach  based  on  the  measurement  of  the  gear  torsional  vibrations  is 
proposed  to  evaluate  the  equivalent  gear  error  of  a  spur  gear  pair  and  to 
identify  the  natural  frequency  and  the  damping  of  the  system.  The  test  bench  is 
modelled  as  a  single  degree  of  freedom  system  and  must  be  realised  by  using 
stiff  bearings  and  torsionally  compliant  shafts.  The  algorithm  is  based  on  the 
use  of  the  harmonic  balance  method.  Results  can  be  obtained  by  using  a  quite 
simple  experimental  apparatus.  The  proposed  approach  has  some  advantages 
with  respect  to  the  traditional  metrological  methods.  The  effect  of 
measurement  errors  on  the  accuracy  of  the  identification  is  also  investigated. 


1.  INTRODUCTION 

In  the  last  decades  many  papers  were  published  on  the  effect  of  gear 
errors  on  the  dynamic  response  of  gear  pairs,  e.g.  references  [1-16].  In  fact 
vibrations  of  gear  pairs  are  largely  affected  by  the  amplitude  and  phase  of 
deviations  of  the  tooth  profile  from  the  true  involute  one.  Pitch,  pressure  angle 
and  mounting  (eccentricities  and  misalignments)  errors  also  are  of  great 
importance.  Therefore  gear  errors  must  be  checked  to  avoid  bad  working 
conditions  of  high  speed  gears  and  silent  reducers.  Moreover  profile 
modifications  are  introduced  to  reduce  gear  vibrations  and  their  accuracy  must 
be  verified.  Analytical  [3,  14,  17-18],  numerical  [2,  9-13,  15,  19-24]  and 
approximate  [6]  methods  were  proposed  in  the  past  to  simulate  the  dynamics 
of  a  spur  gear  pair,  and  single  [2-3,  5-12,  21-22],  multi  [4,  20,  23-24]  or 
infinite  [25]  degrees  of  freedom  were  used  by  different  authors  to  model  the 
system’s  behaviours.  Multi  axes  reducers  were  investigated  e.g.  in  references 
[13-14]. 

In  the  present  study  an  approach  based  on  the  measurement  of  the  gear 
torsional  vibrations  is  proposed  to  evaluate  the  equivalent  gear  error  of  a  spur 
gear  pair  and  to  identify  the  natural  frequency  and  damping  of  the  system.  The 
equivalent  error  is  a  function  of  the  gear  position  and  is  related  to  the  errors  of 
the  driving  and  the  driven  gears  and  to  the  non-dimensional  stiffness  of  the 
teeth;  its  dimension  is  length.  The  system  is  analytically  studied  by  using  a 
single  degree  of  freedom  system  capable  of  modelling  the  experimental  test 


1733 


bench,  which  therefore  must  be  realised  using  stiff  bearings  and  torsionally 
compliant  shafts.  If  gear  pairs  have  different  center  distances,  an  appropriate 
housing  or  different  housings  must  be  built. 

The  algorithm  is  based  on  the  use  of  the  harmonic  balance  method  [26]: 
it  is  applied  to  spur  gear  pairs  having  low  contact  ratio  e{le.,  1  <£<  2)  and  is 
suitable  to  identify  pitch,  profile,  pressure  angle  and  runout  errors.  Results  can 
be  obtained  by  using  a  quite  simple  experimental  apparatus  requiring  only  the 
measurement  of  vibration  response  of  the  driven  gear  during  a  revolution  for 
at  least  three  different  rotational  speeds.  The  gear  pair  must  be  tested  with  a 
fixed  static  load.  When  testing  a  single  gear  this  must  be  coupled  with  a 
reference  gear. 

The  proposed  approach  presents  some  advantages  with  respect  to  the 
metrological  methods  used  to  measure  gear  errors  on  driving  and  driven  gears 
[10].  In  fact  these  methods,  that  uses  control  machines,  provide  the  charts  of 
profile  errors  and  cumulative  pitch  and  runout  errors  for  each  tooth  of  the  two 
gears.  On  the  contrary,  using  the  proposed  technique,  the  equivalent  error  is 
obtained;  it  is  directly  related  to  the  gear  vibrations  and  therefore  is 
particularly  appropriate  to  evaluate  the  gear  accuracy  from  a  dynamic  point  of 
view.  In  fact  it  is  well  known  that  some  modifications  of  the  tooth  involute 
profile  can  provide  a  reduction  of  the  vibration  level,  so  that  the  effect  of  these 
modifications,  the  accuracy  of  gear  profiles  and  mounting  can  be  checked  by 
using  the  equivalent  error. 

The  effect  of  noise  on  the  identification  of  the  equivalent  error  and 
modal  parameters  is  also  investigated.  Some  simulated  tests  are  performed 
with  noise  polluted  vibration  responses  of  the  gear  pair. 

2.  VIBRATION  SIMULATION 

A  pair  of  spur  gears  is  modelled  with  two  disks  coupled  by  nonlinear 
mesh  stiffness,  mesh  damping  and  excitation  due  to  gear  errors.  One  disk  (the 
driving  gear)  has  radius  and  mass  moment  of  inertia /i,  while  the  other  (the 
driven  gear)  has  radius  i?2  and  mass  moment  of  inertia  h',  the  radii  R\  and  R2 
correspond  to  the  radii  of  the  base  circles  of  the  two  gears,  respectively. 

The  transmission  error,  defined  as  the  difference  between  the  actual  and 
ideal  positions  of  the  driven  gear,  is  expressed  as  a  linear  displacement  along 
the  line  of  action.  The  sign  convention  used  for  the  transmission  error  is 
positive  behind  the  ideal  position  of  the  driven  gear.  Analysing  gears  with  low 
contact  ratio  e  (i.e.,  1  <  £  <  2),  the  nonlinear  equation  of  motion  for  the 
dynamic  transmission  error  jc  can  be  written  as; 

mx  +  cjc  +  Z,  (j:,r)  +  /2(A:,r)  =  Wq,  (1) 

where 

J!C=  R,  0,  ^2’  (^) 

being  B\  and  62  the  angular  displacements  of  the  two  gears;  the  equivalent 
inertia  mass  m  of  the  system  is: 


1734 


(3) 


Wo  is  the  static  load  given  by 

W,^TJR,=T,IR^.  (4) 

being  T\  and  T2  the  driving  and  driven  torques,  respectively;  fj  (;c,  f)  are  the 
elastic  forces  of  the  meshing  tooth  pair;,  for;  =  1,2 


when  x-ej(t)>0 
v/hQnx-ej{t)<0. 


(5) 


Obviously,  equation  (1)  can  be  easily  extended  to  high  contact  ratio  gears.  In 
equation  (5)  ki{t)  and  kiit)  are  the  time-varying  meshing  stiffness  of  the  two 
pairs  of  meshing  teeth.  The  error  functions  ei(f)  and  eiit)  are  the  displacement 
excitations  representing  the  relative  gear  errors  of  the  meshing  teeth;  when 
two  pairs  of  teeth  come  into  contact  there  will  be  two  separate  error  functions, 
each  acting  on  a  different  spring.  It  is  assumed  that  positive  error  functions 
give  a  positive  transmission  error.  Error  functions  represent  the  sum  of  pitch, 
profile,  pressure  angle  and  runout  errors.  Moreover,  when  separation  of  tooth 
pairs  occurs,  because  of  the  relative  vibrations  and  backlash  between  the  gear 
teeth,  the  dynamic  forces;^-  (x,  t)  are  zero,  according  to  equation  (5);  these  are 
the  nonlinear  terms  in  the  equation  of  motion.  In  equation  (1)  a  constant 
viscous  damping  is  assumed. 

The  total  stiffness  of  the  gear  pair  is  given  by  k(t)  =  ki{t)+k2{t).  Let  us 
introduce  the  meshing  circular  frequency  co  =  z^2  ,  where  Q  is  the  angular 
velocity  of  the  driven  gear  [rad  s‘‘]  and  z  its  number  of  teeth.  The  stiffness 
kj{t),  which  is  a  periodic  function,  has  a  principal  period  T-27t/co.  The 
behaviour  of  kj{t)  and  k{t)  are  discussed,  e.g.,  in  references  [3,  6,  10]  and  in 
section  4. 

Now  the  case  when  x-ej{t)  >  0  is  considered,  Le.,  when  there  is  contact 
between  the  two  gears.  Therefore  the  following  study  is  correct  when  there  is 
no  tooth  separation  between  driving  and  driven  gears.  The  phenomenon  of 
tooth  separation  is  described,  e.g.,  in  [13].  The  equation  of  motion  (1)  can  then 
be  written  as  a  second  order  linear  ordinary  differential  equation; 

x  +  2^C0qX  +  C0q  K{t)  x=Fq+  cOq  K,  (0  e,  (0  +  col  {t)  (r) ,  (6) 

where:  K{t)  =  k{t)lk„^,  K^{t)-k^{t)lk^,  K^{t)  =  k^{t)lk„^,  Fo=Wo/m  and  f 
is  the  damping  ratio;  K,  K\  and  K2  are  non-dimensional  functions.  Moreover 

the  average  value  of  the  mesh  stiffness  is  =  (i/t)  j/:(r)df  and 

0 

is  the  natural  circular  frequency  of  the  undamped  system  with  stiffness  equal 
to  its  integral  average  value. 

It  is  useful  to  introduce  the  following  Fourier  expansion  of  the 
equivalent  error  [m] 


1735 


(7) 


K,{t)e,(t)  +  K^(f)e^(.t)=  , 

n®-®® 

where  i  is  the  imaginary  unit.  The  expression  on  the  left  side  of  equation  (7) 
represents  the  equivalent  error  of  the  gear  pair;  in  fact,  it  is  the  excitation  due 
to  gear  errors  on  the  right  side  of  equation  (6).  It  is  assumed  that  this  function 
has  a  principal  period  zT,  Le.  the  time  revolution  of  the  driven  gear,  and 
therefore  has  principal  circular  frequency  Q.  Strictly  the  same  tooth  of  the 
driving  and  driven  gears  mesh  together  only  after  a  period  s\zT,  where  and 
$2  are  the  integers  that  express  the  gear  ratio  ras  the  rational  number 
(usually  T  <  1).  In  this  study  the  error  components  having  circular  frequency 
lower  than  Q  are  neglected.  It  is  important  to  observe  that  in  many  cases  the 
profile  errors  can  be  considered  the  same  for  all  the  teeth  of  the  gear,  i.e.  they 
all  have  principal  period  T\  therefore  the  coefficients  dn  for  n  =  z,  2z,  3z, ...  are 
due  to  periodic  profile  errors,  whereas  the  others  are  due  to  pitch,  pressure 
angle  and  runout  errors. 

The  expansion  of  the  non-dimensional  total  meshing  stiffness  is 

a:(i)  .  (8) 

The  solution  of  the  equation  of  motion  (6)  is  obtained  by  using  the  harmonic 
balance  method.  Therefore  the  dynamic  transmission  error  x  is  expanded  into 
a  complex  Fourier  series 

(9) 

n--t» 


Substituting  equations  (7-9)  into  equation  (6)  the  following  equation  is  found 


=  fo+£Bo 


Equation  (10)  gives  the  following  algebraic  linear  system 

AC  =  F,  (11) 

where  the  elements  of  the  matrix  A  are  given  by 

,  (12) 

fl  if  {n-j)/z  isintegerl 

On  /  is  the  Kronecker  delta  and  W„  ^  ,  .  r  •  Moreover  it 

[O  otherwise  J 

is: 


1736 


(13-14) 


0)1 

cold, 

F  =  ^ 

Fo+<old„ 

^-1 

cold., 

.  (old.^  . 

3.  IDENTIFICATION  OF  MODAL  PARAMETERS  AND 
GEAR  ERRORS 

The  aim  of  this  work  is  to  identify  the  equivalent  gear  error;  therefore 
the  vector  F  in  equation  (11)  is  now  unknown.  On  the  contrary  the  contact 
ratio  £,  the  shape  of  the  stiffness  function  Kit)  and  all  the  constants  of  the 
expansion  are  known.  The  transmission  error  x(r)  is  then  experimentally 
measured  for  different  rotational  speeds  Q  of  the  driven  gear;  in  particular, 
only  rotational  speeds  where  no  tooth  separation  occurs  must  be  chosen.  It  is 
obvious  that  the  dynamic  transmission  error  varies  according  to  the  speed  Q, 
so  that  the  constants  c„  of  the  expansion  and  the  vector  C  are  functions  of  O. 
Equation  (12)  shows  that  also  the  matrix  A  is  a  function  of  JQ;  whereas  the 
vector  F  is  independent  of  it.  Therefore  it  can  be  written 

F  =  A(I2)C(I2).  (15) 


In  equation  (15)  the  vector  C(I2)  is  obtained  experimentally  and  the  matrix 
A(I2)  theoretically  by  using  equation  (12).  However  in  order  to  compute  the 
matrix  A  the  modal  parameters  coq  and  f  of  the  system  must  be  identified 
because  they  appear  in  equation  (12).  These  parameters  can  be  determined  by 
using  the  following  equation: 

A  (12, )  C(^2i  )  =  A{£2^)  C(jQ2  )  =  A  (12. )  C(I2,. )  =  constant ,  (16) 


where  A  are  fixed  rotational  speeds.  Then 

,r 


X,  I  '  c,.  +  2 in  f  a)„  c„  +  aj 


=  constant . 


(17) 


Computing  the  left  hand  side  of  equation  (17)  for  different  rotational  speeds, 
e.g.  I2i,  ^22,  123,  and  subtracting  first  the  quantity  computed  for  Q2  to  the  one 
computed  for  Q\  and  then  subtracting  the  quantity  computed  for  to  the  one 
computed  for  one  obtains  the  following  linear  system 


1737 


(18) 


£2nl[r3,  c„  (i3,)-r2,  c„  (A)]  £  X«;[c„.^(i3,)-c„-,(A)] 

^2ri[^3j  C„  (^j)— (^3)]  ^  £*^y  [^n-;;('^l)~^ji-;/(^3)] 


ml  J 


£«^[^2,^c,(^2,)-^23^c„(A)] 

;t=-oo 

%n^[Qfc„(n{)-nic^{Q,)] 


The  linear  system  (18)  allows  to  identify  the  modal  parameters  coq  and  f  and, 
using  equation  (15),  the  vector  F  that  gives  the  equivalent  gear  error. 
Considering  that  usually  col  ^0  is  negligible  with  respect  to  Fq,  also  the  ratio 
Wo/m  can  be  identified;  therefore  if  the  static  load  Wo  is  known,  the  reduced 
mass  m  of  the  system  is  obtained.  Actually  all  the  constant  terms  of  the 
identified  vector  F  can  be  attributed  to  the  static  load,  giving  to  do  the  zero 
value.  In  fact  the  static  load  can  be  easily  considered  as  the  mean  value  of  the 
load  during  the  gear  meshing  and  a  non  zero  coefficient  do  is  equivalent  to 
change  the  static  load. 

In  system  (18)  one  can  substitute  the  quantity  computed  for  the 
difference  Q2-Q1  to  one  of  the  two  differences  previously  computed.  However 
it  is  important  to  observe  that  with  measurements  at  three  different  speeds 
only  two  linearly  independent  equations  can  be  obtained  for  the  system  (18). 

Due  to  the  errors  introduced  in  the  experimental  measurement  of  the 
dynamic  transmission  error  x  (errors  in  C)  it  is  preferable  to  solve  an 
overdetermined  system  using  different  velocities  to  obtain  additional 
equations  in  system  (18).  Moreover  the  problem  is  ill-conditioned,  so  that  the 
errors  on  the  known  vector  C  are  amplified  in  the  solution.  In  order  to 
overcome  this  problem,  it  is  necessary  to  use  only  the  more  significant 
harmonics  in  the  identification  when  significant  measurement  errors  or 
differences  between  the  single  degree  of  freedom  model  and  the  actual  test 
bench  are  observed;  therefore  all  the  sums  involved  in  system  (18)  must  be 
stopped  at  an  integer  n  not  too  large  because  higher  order  harmonics  involved 
in  C  only  introduce  noise  and  does  not  give  additional  information.  This 
process  is  similar  to  the  use  a  low-pass  filter  on  signals  coming  from  sensors 
used  in  experiments.  A  discussion  on  this  phenomenon  is  deferred  to  section 
5.  The  natural  circular  frequency  of  the  system  can  be  evaluated  theoretically 
or  experimentally  and  the  damping  ratio  can  be  also  experimentally 
determined  by  an  impact  test,  using  the  logarithmic  decrement,  or  can  be 
assumed  in  the  range  between  0.07  and  0.1,  as  verified  by  many  authors  [6, 
10].  Therefore  the  results  of  the  identification  can  be  compared  to  data 
obtained  in  a  different  way. 

The  vector  F  can  be  also  determined  by  using  equation  (15)  or  the 
following  expression: 


1738 


F  =  (1//)XA(A)C(A).  (19) 

1=1 

This  procedure  provides  a  good  accuracy  in  the  computation  also  in  presence 
of  measurement  errors,  and  the  average  reduces  the  errors  that  are  introduced 
in  the  experimental  measurement  of  (and  then  in  C). 

4.  APPLICATION  OF  THE  METHOD 

In  the  numerical  simulations  the  stiffness  function  proposed  by  Cai  and 
Hayashi  [6]  was  used;  in  particular  the  following  function  can  be  introduced 

/(f)  =  5;^{^^+((e-l)/2)r]V^[r+((£-l)/2)r]  +  055  .  (20) 

The  two  non-dimensional  meshing  stiffness  K\(t)  and  K2(t)  are  directly 
obtained  by  equation  (20),  i.e.: 

K,{t)  =  fit)  forO<t<T  (21) 


if  i>r-((£-i)/2)7' 
’  [o  if  f<T-((e-l)/2)r 

fit  +  T)  if  r<((£-l)/2)rl  ^ 


if  ?>((£-i)/2)r 


for  0  <  r  <  r. 


The  meshing  stiffness  is  therefore  a  function  of  the  contact  ratio  £.  The  two 
non-dimensional  meshing  stiffness  Ki{t)  and  Kzit)  are  shown  in  Figure  la  (€  = 
1.8)  along  the  meshing  period  T-27t/co,  and  the  non-dimensional  total  meshing 
stiffness  K{t)  is  plotted  in  Figure  lb.  The  integral  average  stiffness  of  the  pair 
k,n  is  related  to  the  maximum  stiffness  of  one  pair  of  teeth  /:max  by  the 
following  expression:  =0.85£  ^max*  ISO/DIS  6336-1.2  (1990)  design 

code  gives  a  formula  to  evaluate  kuAX- 

In  order  to  simplify  the  experimental  measurement  of  the  dynamic 
transmission  error,  only  a  test  on  the  driven  gear  can  be  performed.  Usually 
only  the  acceleration  of  the  driven  gear  ^2  is  measured;  however  the 

acceleration  x  can  be  obtained  by  using  the  following  relationship 

^  .. 

X,  = - X,  (23) 

■  m,  +171;, 

where  mi  and  m2  are  the  reduced  masses  of  the  driving  and  the  driven  gear, 
respectively.  The  measured  acceleration,  can  be  related  to  the  coefficients  c„ 
by  equation  (9),  to  yield: 


1739 


Equation  (24)  shows  that  the  coefficient  cq  cannot  be  obtained  by  vibration 
measurement;  however  this  coefficient  can  be  easily  determined  because  it  is 
the  mean  value  of  the  dynamic  transmission  error  x.  A  good  estimation  of  this 
value  is  Cq  .  In  the  identification  process  it  is  very  important  to  use 

only  measured  accelerations  when  no  tooth  separation  occurs.  Therefore  it  is 
generally  necessary  to  avoid  rotational  speeds  larger  than  half  the  main 
resonance  speed  coq. 

The  benchmark  for  gears  can  be  obtained  by  a  variable  speed  motor  and 
a  brake.  Torsional  flexible  shafts,  with  a  natural  frequency  lower  than  1/10  of 
the  natural  frequency  Q)o  of  the  gear  pair,  and  stiff  bearings  must  be  used  in 
order  to  well  approximate  a  single  degree  of  freedom  system.  The  rotational 
speed  can  be  measured  by  a  proximity  sensor  that  counts  the  number  of 
passing  teeth;  the  load  is  measured  by  a  dynamometer  and  the  acceleration  of 
the  driven  gear  by  accelerometers.  In  particular  two  identical  accelerometers 
which  are  mounted  on  the  gear  through  two  small  aluminium  blocks  are  used; 
they  can  be  placed  on  opposite  sides  of  a  diameter  and  must  have  tangential 
orientation.  A  summing  amplifier  sums  up  the  signals  from  the  accelerometers 
to  eliminate  the  effect  of  eccentricity.  A  slip  ring  is  introduced  to  bring  the 
signals  to  the  amplifier.  In  alternative  a  laser  rotational  vibrometer  can  be 
employed  to  measure  vibrations  of  the  driven  gear;  in  this  case  accelerometers, 
slip  ring  and  summing  amplifier  can  be  eliminated.  Lubrication  of  the  meshing 
teeth  must  be  provided. 


5.  NUMERICAL  RESULTS 

To  verify  the  proposed  method  the  gear  pair  tested  by  Umezawa  et  al. 
[10]  is  studied.  The  two  gears  are  finished  by  a  MAAG  grinding;  the 
characteristics  of  both  the  gears  are:  module  =  4,  number  of  teeth  =  48,  face 
width  =  10  mm,  pressure  angle  =  14.5®,  diameter  of  standard  pitch  circle  = 
192  mm;  the  gear  ratio  is  1.  The  contact  ratio  e  is  1.8,  the  rotational  speed 
range  Q,  -  400  ^  3000  rpm  (41.89  -4-  314.16  rad  s’^),  the  torque  196  Nm  and 
the  teeth  have  involute  profile.  The  profile  errors  are  about  bjim  at  the  root  of 
the  driving  gear  and  at  the  tip  of  driven  gear;  these  profile  errors  are  reported 
in  reference  [10].  The  natural  circular  frequency  is  coo  =  48x3062  rpm  (2450 
Hz)  and  the  value  of  the  damping  ratio  f  is  0.07. 

The  response  of  the  gear  pair  is  polluted  with  noise  on  the  gear  data  and 
modal  parameters.  It  is  assumed  that  only  profile  errors  are  significant  in  this 
case,  so  that  all  the  other  errors  are  neglected.  As  a  consequence  of  this 
hypothesis,  the  dynamic  transmission  error  has  a  principal  period  equal  to  the 
meshing  period  T.  The  response  x  is  discretized  with  201  points  in  the  period. 
Then  the  noise  is  added  to  the  response  before  computing  the  vector  C.  In 
particular  the  noise  is  generated  using  random  numbers  added  to  the  time 
response.  These  random  numbers  are  obtained  by  a  normal  distribution  having 
zero  mean  value  and  variance  G~  lev* max,  where  max  is  the  maximum  value 
of  the  response  x  in  the  period  and  lev  is  the  error  level.  As  a  consequence  of 


1740 


the  assumed  distribution,  the  68  %  of  the  points  have  noise  within  ic7,  the  95 

%  within  ±2(7  and  the  99.7  %  within  ±3(7. 

First,  equation  (18)  is  used  to  identify  the  modal  parameters  of  the 
system  from  noise  polluted  responses.  In  particular  eight  responses  at 
rotational  speeds  Q.  =  60,  65,  68,  70,  80,  90,  100,  120  rad  s*'  with  an  error 
level  /ev  =  0.02  are  employed.  This  error  level  gives  responses  having  a 
difference  within  ±6%  with  respect  to  the  true  value.  In  Figure  2a  the  percent 
difference  between  the  identified  natural  frequency  GOo  and  the  actual  value  is 
plotted  versus  the  number  of  harmonics  used  in  the  identification.  The  range 
of  harmonics  of  the  meshing  frequency  co=z^  that  gives  correct  results  is 
4^<8.  In  Figure  2b  the  data  relative  to  the  damping  ratio  f  are  reported. 
Figure  3  is  similar  to  Figure  2  but  it  is  obtained  for  an  error  level  lev  =  0.005 
(difference  within  ±2.5%  with  respect  to  the  true  value).  In  this  case  the  range 
of  useful  harmonics  increases.  Figure  2  and  3  show  that  it  is  not  convenient  to 
exceed  in  the  number  of  harmonics.  In  fact,  for  the  considered  problem,  the 
amplitude  of  coefficients  c,j  decreases  with  n  so  that  higher  order  harmonics 
are  largely  affected  by  the  noise.  Being  the  identification  process  an  ill 
conditioned  problem,  it  is  necessary  to  employ  only  harmonics  having  a  good 
signal  to  noise  ratio. 

In  Figure  4  the  equivalent  error  obtained  by  using  the  profile  errors 
given  in  reference  [10]  and  equations  (7),  (20-22)  is  shown;  this  one  can  be 
called  the  “actual”  equivalent  error.  In  Figure  5  the  equivalent  error  is 
identified  by  using  a  response  at  =  60  rad  s'^  without  noise.  In  this  case  15 
harmonics  are  used  to  describe  the  function.  The  difference  between  Figures  4 
and  5  can  be  surely  attributed  to  the  truncation  error. 

Then  eight  responses  at  rotational  speeds  speeds  ^2  =  60,  65,  68,  70,  80, 
90,  100,  120  rad  s'‘  with  an  error  level  /ev  =  0.005  are  used  to  identify  the 
equivalent  error;  the  result  is  given  in  Figure  6  where  the  actual  modal 
parameters  (cut)  and  Q  are  used.  Figures  5  and  6  are  very  similar  and  they  well 
describe  the  “actual”  error  reported  in  Figure  4.  Also  in  the  case  of  an  error 
level  of  ^ev  =  0.02  and  using  the  actual  modal  parameters  a  quite  good 
evaluation  of  the  equivalent  error  is  reached,  as  shown  in  Figure  7. 

The  effect  of  an  incorrect  identification  of  the  modal  parameters  coo  and 
^  on  the  evaluation  of  the  equivalent  gear  error  is  then  investigated  in  Figure 
8;  in  this  case  an  error  of  +10%  on  the  frequency  and  +40%  on  the  damping 
ratio  is  used  to  evaluate  the  equivalent  error  by  responses  having  a  noise  level 
lev  ~  0.02 .  Also  in  this  case  a  quite  good  result  is  obtained. 

6.  CONCLUSIONS 

The  reconstruction  of  the  equivalent  gear  error  by  acceleration 
measurement  of  the  driven  gear  of  a  spur  gear  pair  is  quite  good  also  in  the 
case  of  measurements  affected  by  noise.  In  particular  the  identification  of  the 
natural  frequency  and  damping  of  the  system  is  obtained  by  an  overdetermined 


1741 


linear  system  to  minimise  the  error;  to  obtain  correct  results  only  lower  order 
harmonics  must  be  considered.  Then  these  data  can  be  used  to  evaluate  the 
equivalent  gear  error  that  is  not  largely  affected  by  the  inaccuracy  in  the 
identification  of  the  natural  frequency  and  damping. 

The  proposed  method  seems  to  have  advantages  in  the  quality  control  of 
a  large  production  of  gears  having  the  same  center  distance;  in  this  case  the 
same  housing  can  be  used.  Moreover  the  gear  can  be  tested  together  with  a 
reference  gear  or  with  the  companion  gear  to  check  the  actual  gear  pair.  As  a 
limitation,  the  method  can  be  only  applied  to  systems  having  stiff  bearings  and 
torsionally  compliant  axes,  requiring  an  appropriately  designed  experimental 
apparatus. 


REFERENCES 

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383-41 1 .  Mathematical  models  used  in  gear  dynamics. 

2.  H.  N.  OZGUVEN  and  D.  R.  HOUSER  1988  Journal  of  Sound  and  Vibration  125, 
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error. 

3.  M.  AMABILI  and  A.  RiVOLA  to  appear  in  Mechanical  Systems  and  Signal 
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stability  of  the  SDOF  model  with  time-varying  meshing  stiffness. 

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mounting  errors  on  gear  dynamic  behaviour. 

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experiment). 

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vibration  of  power  transmission  spur  gears  (1st  report,  pressure  angle  error  and 
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1743 


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Wiley  and  Sons. 


1.3 
1.2 
1.1 
1 

0.9 
0.8 
0.7 
0,6 
0.5 

0  T 

t 

Figure  1.  (a)  The  non-dimensional  meshing  stiffness  Ki(t)  and  Kzit).  (b)  The 
total  meshing  stiffness  K{t). 


t 


number  of  harmonics 


number  of  harmonics 


Figure  2.  Results  of  the  identification  with  an  error  level  /ev=0.02  (a)  Percent 
difference  between  the  identified  coq  and  the  actual  value  V5.  the  number  of 
harmonics  used  in  the  identification,  (b)  Percent  difference  between  the 
identified  f  and  the  actual  value  v^.  the  number  of  harmonics. 


1744 


difference  [%] 


number  of  harmonics 


number  of  harmonics 


Figure  3.  Results  of  the  identification  with  an  error  level  /ev=0.005  (a)  Percent 
difference  between  the  identified  and  the  actual  value,  (b)  Percent 
difference  between  the  identified  f  and  the  actual  value. 


0 


Figure  4.  Actual  equivalent  gear  error. 


t 


Figure  5.  Identified  equivalent  gear  error  by  response  without  noise. 


1745 


0  T 


t 

Figure  6.  Identified  equivalent  gear  error  by  responses  with 
/5v=:0.005;  actual  modal  parameters. 


0  T 


t 

Figure  7.  Identified  equivalent  gear  error  by  responses  with 
/ev=0.02;  actual  modal  parameters. 


0  T 


Figure  8.  Identified  equivalent  gear  error  by  responses  with 
;ev=0.02;  +10%  of  the  actual  value  of  coo  and  +40%  of  the  actual 


a  noise  level 


a  noise  level 


a  noise  level 
value  of 


1746